$(1)$ Read the "Eigenfaces vs Fisherfaces" paper and answer the following questions:

$(1\mathrm{.}1)[10$ points$]$ When using the Fisherface approach, how to deal with the case when the within$-$class scatter matrix is singular?

$(1\mathrm{.}2)[10$ points$]$ In one of the Eigenface approaches described in the paper, why the first three principal components were removed? Why discarded the first $3$ principal components instead of the first $2$ or $4$ principal components?

$(1\mathrm{.}3)[10$ points$]$ Under what assumptions, the Eigenface approach is equivalent to the correlation method.

$(1\mathrm{.}4)[10$ points$]$ If the training set is representative $($i$.$e$.,$ equally and densely covers all conditions$)$ and the computational resource is not a problem, will the correlation approach outperform the Eigenface and Fisherface approaches?