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COMP3670/6670 Introduction to Machine Learning

创建日期：2024-05-15 17:06:36

所属分类：计算机computer science

标签： COMP3670

Introduction to Machine Learning

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COMP3670/6670 Introduction to Machine Learning
Final Exam
• Write your name and UID on the first page (you will be fine if you forget to write them).
• This is an open book exam. You may bring in any materials including electronic and paper-
based ones. Any calculators (programmable included) are allowed. No communication devices
are permitted during the exam.
• Reading time: 30 minutes
• Writing time: 180 minutes
• For all the questions, write your answer CLEARLY on papers prepared by yourself.
• There are totally 9 pages (including the cover page)
• Points possible: 100
• This is not a hurdle.
• When you are asked to provide a justification to your answer, if your justification is incorrect,
you will get 0.
1
• Section 1. Linear Algebra and Matrix Decomposition (13 points)
1. (6 points) Show that it is impossible to have a set of 3 orthogonal non-zero vectors
{v1,v2,v3} in R2.
(Hint: Write v1 = [x1, y1]
T and first assume that x1 = 0. Then, prove it for the case where
x1 ̸= 0, using the first case to help you.)
Solution. We write
v1 =
[
x1
y1
]
v2 =
[
x2
y2
]
v3 =
[
x3
y3
]
As the hint suggests, we first assume that x1 = 0. Then since v1 ·v2 = 0, we have y1y2 = 0.
Since y1 cannot be zero (otherwise v1 = 0) we must have y2 = 0. Since v1 · v3 = 0,
we have x1x3 + y1y3 = y1y3 = 0, and y1 ̸= 0, so y3 = 0. Since v2 · v3 = 0, we have
x2x3 + y2y3 = x2x3 = 0. Now, y2 = 0, so x2 ̸= 0 (else v2 = 0), so x3 = 0. We have that
x3 = 0 and y3 = 0, so v3 = 0, a contradiction.
Now, if any of the x1, x2, x3, y1, y2, y3 were zero, this would also result in a contradiction
in the same way, as by symmetry we can reorder the vectors so the zero was somewhere in
the first vector, and since x1y1 + x2y2 = y1x1 + y2x2, we could swap the x and y values in
all vectors so that x1 = 0, from which the contradiction follows by above.
Now, assume that x1 ̸= 0. v1 · v2 gives
v1 · v2 = 0
x1x2 + y1y2 = 0
x2 =
−y1y2
x1
and similarly
v1 · v3 = 0
x1x3 + y1y3 = 0
x3 =
−y1y3
x1
Then v2 · v3 = 0, so
x2x3 + y2y3 = 0(−y1y2
x1
)(−y1y3
x1
)
+ y2y3 = 0
y21y2y3
x21
+ y2y3 = 0
y2y3
(
y21
x21
+ 1
)
= 0
Then either y2 = 0 (contradiction), or y3 = 0 (contradiction) or
y21
x21
+1 = 0, giving y21 = −x21.
Since the left hand side is non-negative, and the right hand side is non-positive, the only
way to satisfy this equation is y1 = x1 = 0, a contradiction.
2. (7 points) Consider an arbitrary matrix A ∈ R2×2
A =
[
a b
c d
]
We would like A to satisfy the following properties:
(a) A is non-invertible.
(b) A is symmetric.
2
(c) All the entries of A are positive.
(d) A has a positive eigenvalue λ = p > 0.
Find the set of all matrices A (as a function of p) that satisfy all the above constraints.
Solution. A is symmetric, so b = c. We can eliminate c and consider only matricies of the
form
A =
[
a b
b d
]
.
A is non-invertible, so ad − b2 = 0. Also, a, b, d > 0. We have that p > 0 is an eigenvalue,
so we find all eigenvalues using the characteristic polynomial.
(a− λ)(d− λ)− b2 = 0
ad− aλ− dλ+ λ2 − b2 = 0
−aλ− dλ+ λ2 = 0
λ(λ− a− d) = 0
So λ = 0 or λ = a+ d. Since 0 cannot be positive, we have that p = a+ d, or d = p− a. We
can eliminate d.
A =
[
a b
b p− a
]
Finally, since ad − b2 = 0, we have a(p − a) = b2, and hence b = ±√a(p− a). Since all
entries need to be positive, we take the positive root, b =
√
a(p− a). For the square root
to be defined (and not zero), we need a(p − a) > 0, which implies that 0 < a < p. Hence,
the set of all possible A is given by{[
a
√
a(p− a)√
a(p− a) p− a
]
: 0 < a < p
}
as required.
3
• Section 2. Analytic Geometry and Vector Calculus (12 points)
1. (6 points) Find all matrices T ∈ R2×2 such that for any v ∈ R2,
vTv = (Tv)Tv = (Tv)TTv
Solution. Consider v · v = T (v) · v. Let v = [x, y]T . Then
v · v = T (v) · v
x2 + y2 = x(ax+ by) + y(cx+ dy)
x2 + y2 = ax2 + (b+ c)xy + dy2
By choosing x = 0, y = 1, we obtain d = 1, and by choosing x = 1, y = 0 we obtain a = 1.
Then, we choose x = 1 and leave y arbitrary for the moment. Substituting, we obtain
1 + y2 = 1 + (b+ c)y + y2
from which we obtain (b+ c)y = 0 for all y, so c = −b. We now consider the other equation
v · v = T (v) · T (v)
x2 + y2 = (ax+ by)2 + (cx+ dy)2
x2 + y2 = (a2 + c2)x2 + (b2 + d2)y2 + 2(ab+ cd)xy
x2 + y2 = (1 + b2)x2 + (1 + b2)y2 + 2(b− b)xy
x2 + y2 = (1 + b2)(x2 + y2)
1 = 1 + b2
b = 0
So a = 1, b = 0, c = −b = 0, d = 1. So the only matrix that satisfies this property is the
identity.
T =
[
1 0
0 1
]
2. (6 points) Let x,a ∈ Rn×1, and define f : Rn×1 → R as
f(x) = 3 exp(2xTAx)
where exp(x) := ex, the exponential function. Compute ∇xf(x).
Solution. We apply the chain rule. Note that f(x) = g(h(x)), where g : R → R, g(h) =
3 exp(2h) and h : Rn → R, h(x) = xTAx. We have by lectures/assignment that ∇xh(x) =
xT (A+AT ), and by calculus we have that ∇hg(h) = 6 exp(2h). Therefore
df
dx
=
dg
dh
dh
dx
= 6 exp(2h)xT (A+AT ) = 6 exp 2(xTAx)xT (A+AT )
4
• Section 3. Probability (15 points)
Consider the following scenario. I have two boxes, box A and box B. In their initial configuration,
Box A contains 3 red balls and 3 white balls; Box B contains 6 red balls.
We swap the balls in boxes via the following procedure.
Procedure 1:We draw a ball xA uniformly at random from box A, and draw a ball xB uniformly
at random from box B. We place ball xA into box B and place ball xB into box A.
Let RA and RB be random variables representing the number of red balls in box A and box B
respectively.
1. (3 points) Describe the probability mass functions p(Ra = n) and p(Rb = n) for all integers
n ≥ 0 for the initial configuration of the boxes.
Solution. Clearly, box A contains 3 red balls, so
p(RA = n) =
{
1 n = 3
0 n ̸= 3
and box B contains 6 red balls, so
p(RB = n) =
{
1 n = 6
0 n ̸= 6
2. (3 points) After performing Procedure 1, what is the new joint probability mass functions
p(RA = na, RB = bn)?
Solution. We are guaranteed to move a red ball from box B into box A, it is equally likely
that the ball xA chosen from A is red. So, we have a 0.5 probability of nothing changing.
and a probability of 0.5 of adding a red ball to box A and loosing a red ball from box B.
Hence,
p(RA = na, RB = nb) =

0.5 (na, nb) = (3, 6)
0.5 (na, nb) = (4, 5)
0 else

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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