辅导案例-COMP3670-Assignment 2

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The Australian National University Semester 2, 2020
Research School of Computer Science Theory Assignment 2 of 5
COMP3670: Introduction to Machine Learning
Release Date. Aug 21st, 2020
Due Date. 23:59pm, Sep 13th, 2020
Maximum credit. 100
Exercise 1 Orthogonal Compliments (10+10 credits)
Let V be a vector space, together with an inner product 〈·, ·〉 : V × V → R and let X and Y be vector
subspaces of V . We define the orthogonal compliment XT as
XT :=
{
v ∈ V : 〈x,v〉 = 0 for all x ∈ X}
1. Prove that X ∩XT = {0}, where 0 is the zero vector in V .
2. Prove that if X ⊆ Y , then Y T ⊆ XT .
Exercise 2 Norms and Inner Products (10+20 credits)
1. Let (V, 〈·, ·〉) be an inner product space. Let
proju(v) :=
〈v,u〉
〈u,u〉u
denote the vector projection of v onto u. Prove that v− proju(v) and u are orthogonal.
2. Let (V, 〈·, ·〉) be an inner product space. Let ||x|| := √〈x,x〉. Prove that || · || is a norm.
(Hint: To prove the triangle inequality holds, you may need the Cauchy-Schwartz inequality, 〈x,y〉 ≤
||x||||y||.)
Exercise 3 Vector Calculus (10+10+30 credits)
1.
f, g : Rn → R, f(x) = cTx, c ∈ Rn, g(x) =

cTx+ µ2, µ ∈ R.
• a) (3 points) Prove df(x)dx = cT .
• b) (2 points) Calculate dgdx .
2. Given a system of linear equations Ax = b, with A ∈ Rk×n,x ∈ Rn×1, b ∈ Rk×1, sometimes there
exists no solutions x. So we’d like to find a approximate solution Ax ≈ b. To achieve this, we
formulate the following regularized least squares error
`(x) =‖Ax− b‖22 + λ‖x‖22 ,where λ ∈
Show that the gradient of the regularized least squares error above is given by
d`(x)
dx
= 2(xTATA− bTA) + 2λxT
(Hint: you can directly use the conclusions from questions 2 and 3 above, together with the definition
of the Euclidean norm.)
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