Questions on data structure suitability

[dkioroglou]

Hello,

I'd like to implement SGL, but I have some doubts on whether it could be applied on the data structure I have.
From my understanding, to implement SGL the features in $X$ should form blocks of groups.
In my case I have the following:

• $n$ samples that come from 5 different groups.
• $p$ features that are the same for all groups.
• $X$ with shape $n \times p$.
• a continuous $y$ outcome.

The research question is to select features that are associated with $y$ regardless of the group (=main effects), and features that are specific to a group (=group effects).

Since the samples form groups and not the features, I thought of creating interactions between $X$ and the groups:
$$X_{gj} = X \cdot I(group=j)$$

and expand $X$ with the interactions:
$$[X,X_{g1},X_{g2}, X_{g3}, X_{g4}, X_{g5}]$$

Based on the above, I have the following questions:

1. Does the expansion of $X$ with interactions make sense for SGL?
2. Since interactions create sparse rows, will that affect negatively SGL?

Thank you in advance.