$($a$)(5$ points$)$ What are the k projection directions chosen by PCA? Why are these directions chosen? $($Be sure to describe the objective of the PCA optimization problem in words.$)($b$)(5$ points$)$ What is the difference between parametric and non$-$parametric density estimation? Describe a benefit and disadvantage of each. $($c$)(5$ points$)$ What convex optimization problem do we solve when we train a logistic regression classifier? Give the mathematical formulation and describe what we are optimizing in words.