## Description

1. Have you read through the class syllabus, noted the important dates, and the class policies?

2. (i) Which of the following courses have you taken?

• CSci 5512 – Artificial Intelligence II

• CSci 5521 – Introduction to Machine Learning

• CSci 5523 – Introduction to Data Mining

(ii) Have you taken any course on Probability/Statistics? If yes, please write down the

course department and course name.

(iii) Have you taken any course on Linear Algebra? If yes, please write down the course

department and course name.

(iv) Have you taken any course on Optimization? If yes, please write down the course

department and course name.

3. Let X ∈ R

n×p and y ∈ R

n be given. The goal is to find a w

∗ ∈ R

p which solves the following

problem:

min

w∈Rp

1

2

ky − Xwk

2 +

c

2

kwk

2

,

where c > 0 is a constant. Give a closed form expression for w

∗

in terms of X, y and c. (Consult

the Matrix Cookbook if you want to look up expressions for derivatives in matrix/vector form.)

4. Let A be a n × n positive definite matrix. The solutions to the following problems

max

w∈Rn:wT w≤1

w

T Aw and min

w∈Rn:wT w≤1

w

T Aw (1)

have well known names—do you know what the solutions to these problems are called? (You

can refer back to your Linear Algebra course if needed)

5. What is the probability density function p(x; µ, Σ) of a multivariate Gaussian distribution

with mean µ and covariance Σ? Please provide an expression in terms of x, µ, Σ, and clearly

define any special function you use in the expression.

Let Θ = Σ−1 be the precision or inverse covariance matrix. What is expression of the

probability density function p(x; µ, Θ−1

) of a multivariate Gaussian distribution in terms of

the mean µ and precision matrix Θ?