Finding a regularized common projection in model-based clustering (…or so I thought)

Suppose that a finite mixture model identified 3 components from a data set. Which variables contributed the most in identifying said components? That was the initial question. In particular, I wanted to find a smaller, rather than larger, number of important variables, since that would make the investigator’s life easier. Contrary to my expectation, the first-devised strategy was not satisfactory, so I had to backtrack a little after much confusion. On that note, I will discuss briefly the problem of finding a regularized common projection for a finite mixture model, and share my experience in backtracking and re-evaluating my approach.