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COMP3670/6670: Introduction to Machine Learning
创建日期：2024-06-03 15:07:20
标签： COMP3670
Introduction to Machine Learning

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COMP3670/6670: Introduction to Machine Learning
Exercise 1 Orthogonal Projections (3 + 3 + 3 + 4 + 6 + 3 credits)
Consider the Euclidean vector space R3 with the dot product. A subspace U ⊂ R3 and vector x ∈ R3 are
given by:
U = span


−1
1
1
 ,

2
−1
−2

 ,x =

8
4
16

1. Show that x ̸∈ U .
2. Determine the orthogonal projection of x onto U , denoted πU (x).
3. Determine the distance d(x, U) := miny∈U ∥x− y∥, where ∥ · ∥ denotes the Euclidean norm.
4. Use Gram-Schmidt orthogonalization to transform the matrix A =

−1 2
1 −1
1 −2
 into a matrix B
with orthonormal columns.
5. Let Q ∈ Rm×n be a matrix with orthonormal columns and x ∈ Rm be an m-dimensional vector.
Find the vector θ that minimizes ||x−Qθ||2 + λ||θ||2, where λ is a positive real number.
6. Compute the vector θ for the matrix B and λ = 10.
Exercise 2 Vector calculus practices (6 + 8 + 8 credits)
Compute the following gradients over x or X. Represent the result in numerator layout. Note that you
are only allowed to use the rules demonstrated in the lecture. Show each step clearly.
1.
∂xTABC x
∂x
2.
∂(Bx+ b)
T
C(Dx+ d)
∂x
3.
∂ tr(X2)
∂X
Exercise 3 Concavity of a function (8 + 10 + 10 credits)
A function f : Rn → R with a convex domain is called a concave function if and only if its Hessian
H = ∂
2f
∂x2 is negative semidefinite. Consider the following function:
f(x) =
(
n∑
i=1
xpi
)1/p
with convex domain dom(f) = Rn++ (n-dim strictly elementwise positive vectors), and p < 1, p ̸= 0.
1. Evaluate the elementwise second order derivatives ∂
2f
∂xixj
for arbitrary integer i, j ∈ [1, n].
2. Denote the elementwise power of a vector a ∈ Rn++ to a real number t as at =[
a1
t a2
t · · · ant
]T
. Also, the diag(·) function returns the diagonal matrix with diagonal values
input as a vector. Prove that
H = (1− p)f(x)1−2p ·
(
xp−1 · xp−1T − f(x)p · diag (xp−2))
3. Prove H is negative semidefinite, hence f is concave since it has a convex domain.
Exercise 4 Expectations with respect to a Gaussian distribution (10+10+8 credits)
A common objective function in modern machine learning is the variational free-energy,
F(q(θ)) =
∫
dθq(θ) log
q(θ)
p(θ)p(y|θ, x) =
∫
dθq(θ)[log q(θ)− log p(θ)− log p(y|θ, x)]. (1)
Consider a simplified setting in which
p(θ) = N (θ; 0, 1), (2)
p(y|θ, x) = N (y; θx, σ2n), (3)
q(θ) = N (θ;µ, σ2), (4)
where N (x;m, v) means x is a univariate Gaussian random variable with mean m and variance v.
1. Compute F .
2. Find the gradients ∂∂µF and ∂∂σF .
3. Set these gradients to zero and solve for µ and σ in terms of y, x and σn.
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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 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