text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
to show that the cardinality of this set is the same as the cardinality of the set
that interests us). In this lecture, this set might be uncountable. Therefore, we need to
introduce a metric on this set so that we can treat the close points in the same manner. To
this end we will define covering numbers (which basicall... | https://ocw.mit.edu/courses/18-657-mathematics-of-machine-learning-fall-2015/5ebb42429b252cbd2f711cd03e01b97f_MIT18_657F15_L6.pdf |
the empirical l1 distance as
to be ℓ
F
F
◦ F
dx
1(f, g) =
1
n
n
i
=1
X
f (x )
i
|
.
g(xi)
|
−
Theorem: If 0
f
≤
≤
1 for all f
∈ F
, then for any x = (x1, . . . , xn), we have
ˆRx
n(
F
)
inf
ε≥0
≤
ε +
(cid:8)
2 log (2N (
F
n
r
x
1 , ε))
, d
.
(cid:9)
Proof. Fix x = (x1, . . . , xn) and ε > 0. Let V be a minimal ε-net of... | https://ocw.mit.edu/courses/18-657-mathematics-of-machine-learning-fall-2015/5ebb42429b252cbd2f711cd03e01b97f_MIT18_657F15_L6.pdf |
inf ε +
ε≥0
(cid:8)
2 log(2N (
F
n
r
, dx
1 , ε))
.
(cid:9)
The previous bound clearly establishes a trade-off because as ε decreases N (
creases.
F
, dx
1 , ε) in-
5.2.2 Computing Covering Numbers
As a warm-up, we will compute the covering number of the ℓ2 ball of radius 1 in dR denoted
ε )d. There are several techniqu... | https://ocw.mit.edu/courses/18-657-mathematics-of-machine-learning-fall-2015/5ebb42429b252cbd2f711cd03e01b97f_MIT18_657F15_L6.pdf |
2) ,
ε
2
V
|
| ≤ (cid:0)
1 + ε d
2
d
ε
2
(cid:1)
=
2
ε
(cid:18)
d
+ 1
(cid:19)
≤
(cid:18)
3 d
ε
(cid:19)
.
(cid:0)
(cid:1)
6
For any p
1, define
≥
and for p =
, define
∞
dx
p(f, g) =
f (xi)
|
−
g(x ) p
i
| !
1
p
,
1
n
n
i=1
X
dx
∞(f, g) = max
i
f (xi)
|
−
.
g(xi)
|
Using the previous theorem, in order to bound
with dx
n... | https://ocw.mit.edu/courses/18-657-mathematics-of-machine-learning-fall-2015/5ebb42429b252cbd2f711cd03e01b97f_MIT18_657F15_L6.pdf |
(f, dx
. Using H¨older’s inequality with r = q
∞, ε)
⊆
p
≤
≤
∞
≥
1 we obtain
p ≥
N (f, dp, ε). Now suppose that
1
p
z
i
|
p
|
!
1
n
n
i=1
X
1
−n p
≤
n
(1
−
1
1
)
r p
n
1
!
i 1
=
X
zi
|
pr
|
!
i=1
X
1
pr
=
1
q
.
zi
|
q
|
!
1
n
n
i
=1
X
This inequality yeilds
B(f, dx
q , ε) =
g : dx
{
q (f, g)
ε
} ⊆
≤
B(f, dx
p, ε),
whic... | https://ocw.mit.edu/courses/18-657-mathematics-of-machine-learning-fall-2015/5ebb42429b252cbd2f711cd03e01b97f_MIT18_657F15_L6.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.641 Electromagnetic Fields, Forces, and Motion, Spring 2005
Please use the following citation format:
Markus Zahn, 6.641 Electromagnetic Fields, Forces, and Motion, Spring
2005. (Massachusetts Institute of Technology: MIT OpenCourseWare).
http://ocw.mit.edu (accessed MM DD, ... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/5ee93bfd5358d2f6a8da82ad8874c029_lecture2.pdf |
, z dxdy
)
3'
1
3
Φ ≈ ∆ ∆ ∆
⎧
⎪
x y z
⎨
⎪
⎩
(
⎡A x, y, z
⎣ x
)
( - ∆x, y, z
− A x
x
∆x
)⎤
⎦
+
⎣ y (
⎡A x, y + ∆y, z
y (
− A x, y, z
)⎦
⎤
)
∆y
+
⎡
A x, y, z + z
⎣ z
(
z
∆ −) A x, y, z
(
⎤ ⎫
⎦ ⎪
⎬
∆z
⎪⎭
)
≈ ∆V ⎢
⎡ ∂Ax
⎣ ∂x
+
∂Ay
∂y
+
∂Az ⎤
⎥
∂z ⎦
(cid:118)∫ A dS
i
=
div A = lim S
V 0 ∆V ... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/5ee93bfd5358d2f6a8da82ad8874c029_lecture2.pdf |
(cid:118)∫ A dS = ∑ (cid:118)∫
i
A dS
i
i
S
i=1
N→∞
dS
i
N
= lim ∑ (∇ i A) ∆Vi
N→∞
V 0
∆ →
n
i=1
= ∫ ∇ i A
V
dV
∫V
∇ i A dV = (cid:118)∫ S
A i da
3. Gauss’ Law in Differential Form
(cid:118)∫ ε0 E i da = ∫ ∇ i (ε0E dV =
)
∫ ρ dV
S
V
V
∇ i ε0E = ρ
)
(
µ H i da = ∇ i µ H dV = 0
( 0 )
∫
V
0
S
... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/5ee93bfd5358d2f6a8da82ad8874c029_lecture2.pdf |
∂x ⎦
⎥ + i
−
z ⎢
⎡ ∂Ay
⎣ ∂x
-
∂A ⎤
x ⎥
∂y ⎦
= det ⎢
−
⎡ −
− ⎤
i z ⎥
i y
⎢ i x
∂ ⎥
⎢ ∂
∂
⎥
∂z ⎥
⎢ ∂x
∂y
⎥
⎢
⎢Ax Ay Az ⎥
⎦
⎣
= ∇ × A
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 2
Page 6 of 10
2. Stokes’ Integral Theorem
Courtesy of Krieger Publishing. Use... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/5ee93bfd5358d2f6a8da82ad8874c029_lecture2.pdf |
i da
S
∇ × H = J + ε0
∂E
∂t
III. Applications to Maxwell’s Equations
1. Vector Identity
lim A i ds = 0 = ∇ × A i da = ∇ i ∇ × A dV
C 0→
∫
(cid:118)∫
(
)
)
C
V
(cid:118)∫ (
S
∇ i (∇ × A ) = 0
2. Charge Conservation
⎧
⎪
∇ × H = J + ε0
∇ i ⎨
⎪
⎩
⎫
∂E
⎪
⎬
∂t ⎪⎭
0 = ∇ i
⎡
⎢J + ε0
⎢
⎣
⎤
∂E
⎥
∂t ⎥
... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/5ee93bfd5358d2f6a8da82ad8874c029_lecture2.pdf |
⎡
J + ε0
∇ i
⎢
⎢⎣
⎤
E
∂
⎥
t
∂
⎥⎦
= 0
MQS Limit
∂H
∇ × E = −µ 0 ∂t
∇ i E = −∇ i ∇Φ = −∇ Φ =
2
(
)
(Poisson’s Eq.)
∇ × H = J
ρ
ε 0
Φ x, y, z =
)
(
∫∫∫
x ',y ',z ' 4πε0 ⎣
ρ (x ', y ', z ') dx ' dy ' dz '
+ (y − y ')2
⎡(x − x ')2
1
+ (z − z ')2
⎤
⎦
2
∇ µ0
i ( H) = 0 ⇒ µ H = ∇ × A
0
∇ 2 A = − µ 0 ... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/5ee93bfd5358d2f6a8da82ad8874c029_lecture2.pdf |
3.46 PHOTONIC MATERIALS AND DEVICES
Lecture 1: Optical Materials Design Part 1
Lecture
Notes
Goal: To develop principles for optical
materials design.
Approach:
Physical basis of properties;
use properties in design.
Electromagnetic Field
Apply voltage: E = ( ,r t
)
Apply current: H = ( ,r t
)
K
K
K
K
Maxwe... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/5efbe006f817e26278e1d29903eed191_3_46lec1_optmat1.pdf |
ittivity of medium
ε
ε0
= dielectric constant
(static)
ε
ε
0
11.7
16
43
3600
Si
Ge
LiNbO3
BaTiO3
Static:
ν =
0
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 1: Optical Materials Design Part 1
Page 2 of 6
Notes
Lecture
2... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/5efbe006f817e26278e1d29903eed191_3_46lec1_optmat1.pdf |
σλ
mW
nm
Power
Spectral Bandwidth
Fiber
dB/km
α
σ τ /L ns/km
L
km
Attenuation
Response Time
Length
Detector
photons/bit Sensitivity
Data Rate
bits/s
n0
B0
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 1: Optical Materials Design Part 1
Page 4 of 6
... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/5efbe006f817e26278e1d29903eed191_3_46lec1_optmat1.pdf |
1
dt 2
dt
3
accel.
vel.
x (position)
Linear differential equation
Resonances
Driven simple harmonic oscillators
K
2d P
dt 2
K
K
2P + ω ε χ E
2
0
K
dP
dt
= −σ
− ω0
0
0
K
K
K
) = ε χ ν )E
P = N ex
0
(
(
Dipole movement
# charges/unit volume
K
P
= ε
0
⎤
⎡
K
χ ω 2
⎥
⎢
0
0
E
⎢ (ω − ω ) − jσω ⎥
2
... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/5efbe006f817e26278e1d29903eed191_3_46lec1_optmat1.pdf |
Ten Lectures and Forty-Two Open Problems in the Mathematics of
Data Science
Afonso S. Bandeira
December, 2015
Preface
These are notes from a course I gave at MIT on the Fall of 2015 entitled: “18.S096: Topics in
Mathematics of Data Science”. These notes are not in final form and will be continuously
edited and/or correc... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
. . . . . . . . . . . . . . . . . . . .
0.2.2 Matrix AM-GM inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.3 Brief Review of some linear algebra tools . . . . . . . . . . . . . . . . . . . . . . . . . .
Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spectral ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
1.2.1 A related open problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Spike Models and BBP transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
. . . . . . . .
2.2.3 A simple example
. . . . . . . . . . . . . . .
2.2.4
2.3 Semi-supervised learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 An interesting experience and the Sobolev Embedding Theorem . . . . . . . .
Similar non-linear dimensional reduction techniques
3 Spectral C... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
. . . . . . . . . . . . . . . . . . . . . . . . .
4 Concentration Inequalities, Scalar and Matrix Versions
4.1.1
4.1 Large Deviation Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sums of independent random variables . . . . . . . . . . . . . . . . . . . . . . .
4.2 Gaussian Concentratio... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Optimality of matrix concentration result for gaussian series . . . . . . . . . . . . . . .
63
63
63
63
64
66
4.5.1 An interesting observation regarding random matrices with independent matrices 68
69
69
70
75
75
76
77
. . . . . . . . . . . . . . . .
4.6.1 A... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
. . . . . . .
5.2 Gordon’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Gordon’s Escape Through a Mesh Theorem . . . . . . . . . . . . . . . . . . . .
5.2.2 Proof of Gordon’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
. . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.2 Equiangular Tight Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.3 The Paley ETF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 The Kadison-Singer problem . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
3.2 The deletion channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Approximation Algorithms and Max-Cut
108
8.1 The Max-Cut problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2 Can αGW be improved? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
. . . . . . . . . . . . . . . . . . . . 119
9.4 Exact recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.5 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.6 The analysis . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
. . . . . . . . . . . . . . . . . . . . . . 129
9.13 Another conjectured instance of tightness
Some preliminary definitions
9.6.1
10 Synchronization Problems and Alignment
131
10.1 Synchronization-type problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
10.2 Angular Synchronization . . . . . . . .... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
[MS15].
• 2.1: Ramsey numbers
• 2.2: Erdos-Hajnal Conjecture
• 2.3: Planted Clique Problems
• 3.1: Optimality of Cheeger’s inequality
• 3.2: Certifying positive-semidefiniteness
• 3.3: Multy-way Cheeger’s inequality
• 4.1: Non-commutative Khintchine improvement
• 4.2: Latala-Riemer-Schutt Problem
• 4.3: Matrix Six devia... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
Positive PCA tightness
• 10.1: Angular Synchronization via Projected Power Method
• 10.2: Sharp tightness of the Angular Synchronization SDP
• 10.3: Tightness of the Multireference Alignment SDP
• 10.4: Consistency and sample complexity of Multireference Alignment
0.2 A couple of Open Problems
We start with a couple of... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
the fol-
lowing is true:
(a)
and
(b)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
1
n!
(cid:88)
n
(cid:89)
Aσ(j)
σ∈Sym(n)
j=1
1
n!
(cid:88)
σ∈Sym(n)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
n
(cid:89)
Aσ(j)
j=1
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:1... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
fly review a few linear algebra tools that will be important during the course.
If you need a refresh on any of these concepts, I recommend taking a look at [HJ85] and/or [Gol96].
0.3.1 Singular Value Decomposition
The Singular Value Decomposition (SVD) is one of the most useful tools for this course! Given a
matrix M ∈... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
|
(cid:107)M (cid:107) = max |λk(M ) .
k
0.3.3 Trace and norm
Given a matrix M ∈ Rn×n, its trace is given by
Tr(M ) =
n
(cid:88)
k=1
Mkk =
n
(cid:88)
k=1
λk (M ) .
Its Frobeniues norm is given by
(cid:107)M (cid:107)F =
(cid:115)(cid:88)
ij
M 2
ij = Tr(M T M )
A particularly important property of the trace is that:
Tr(... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
that (2) is maximized by taking v1, . . . , vd to be the k leading eigenvectors of M
and that its value is simply the sum of the k largest eigenvalues of M . The nice consequence of this
is that the solution to (2) can be computed sequentially: we can first solve for d = 1, computing v1,
then v2, and so on.
Remark 0.2 A... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
of unknown variables from pair-
wise ratios on compact groups.
11. Some extra material may be added, depending on time available.
0.4 Open Problems
A couple of open problems will be presented at the end of most lectures. They won’t necessarily
be the most important problems in the field (although some will be rather imp... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
as
and its sample covariance as
µn =
n
1 (cid:88)
n
k=1
xk,
Σn =
1
−
1
n
n
(cid:88)
k=1
xk
(
− µn)
(xk
−
Tµ
n) .
(4)
(5)
Remark 1.1 If x1, . . . , xn are independently sampled from a distribution, µn and Σn are unbiased
estimators for, respectively, the mean and covariance of the distribution.
We will start with the fir... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
��es
(cid:33)
xk − nµ∗ − V
(cid:32)
n
(cid:88)
k=1
(cid:32)
n
(cid:88)
(cid:33)
βk = 0.
k=1
Because (cid:80)n
k=1 βk = 0 we have that the optimal µ is given by
µ∗ =
n1
(cid:88)
n
k=1
xk = µn,
the sample mean.
We can then proceed on finding the solution for (9) by solving
n
(cid:88)
k=1
min
V, βk
V T
V =I
(cid:107)x −
k
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
:13)
2
(cid:13) = (xk
2
T
− µn) (xk − µn)
Tµn) V V T
−2 (xk −
T
+ (x − µ ) V V T V V T
Tµn) (xk − µn)
(x
k
(cid:1)
(cid:0)
n
k
= (xk −
µ
− n)
(xk − µn)
Since (xk −
Tµn) (xk − µn) does not depend on V , minimizing (9) is equivalent to
− (xk −
Tµn) V V T (xk − µn) .
max
V T V =I
n
(cid:88)
k=1
(xk − µn) V V T (xk − µn) .... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
rst show that interpretation (2) of finding the d-dimensional projection of x1, . . . , xn that
preserves the most variance also arrives to the optimization problem (13).
1.1.2 PCA as d-dimensional projection that preserves the most variance
We aim to find an orthonormal basis v1, . . . , v
d (organized as V = [v1, . . .... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
(13) and
that the
two interpretations
of
PCA
are indeed equivalent.
1.1.3 Finding the Principal Components
When given a dataset x1, . . . , xn ∈ Rp, in order to compute the Principal Components one needs to
find the leading eigenvectors of
Σn =
1
−
1
n
n
(cid:88)
k=1
(
xk
− n) (xk − µn) .
µ
T
A naive way of doing this w... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
:1) time
are randomized algorithms that compute an approximate solution in
1
(see for example [HMT09, RST09, MM15]).
T
1.1.4 Which d should we pick?
Given a dataset, if the objective is to visualize it then picking d = 2 or d = 3 might make the
most sense. However, PCA is useful for many other purposes, for example: (1... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
any orthonormal basis V = [v1, . . . , vp]
of Rp, consider the following random variable ΓV : Given a draw of the random vector g, ΓV is the
squared (cid:96)2 norm of the largest projection of g on a subspace generated by d elements of the basis V .
The question is:
What is the basis V for which E [ΓV ] is maximized?
2... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
)
(cid:0)
vT
i g
i∈S
2
(cid:1)
,
where g ∼ N (0, Σ). The observation regarding the different ordering of the steps amounts to saying
that the eigenbasis of Σ is the optimal solution for
(cid:34)
argmax max E (cid:88) (cid:0)vT
V ∈Rp×
p
V T V =I
S⊂[p]
|
=d
S|
∈S
i
i g(cid:1)2
(cid:35)
.
1.2 PCA in hi... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
For simplicity we will instead try to understand the spectral properties of
1
Sn = XX T .
n
Since x ∼ N (0, Σ) we know that µn → 0 (and, clearly, n
essentially the same as Σn.3
n−1 → 1) the spectral properties of Sn will be
Let us start by looking into a simple example, Σ = I. In that case, the distribution has no low
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
is plotted as the red line in the figure above.
−
Remark 1.2 We will not show the proof of the Marchenko-Pastur Theorem here (you can see, for
example, [Bai99] for several different proofs of it), but an approach to a proof is using the so-called
moment method. The core of the idea is to note that one can compute moments... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
(n) defined over the complex numbers, meaning simply
that each entry of X is an iid complex valued standard gaussian CN (0, 1) the reverse inequality is
conjectured for all n ≥ 1:
Notice that the singular values of √1 X are simply the square roots of the eigenvalues of Sn,
n
αC(n + 1) ≤ αC(n).
(cid:19)
(cid:18) 1
√ X
n
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
+ βvvT ,
for v a unit norm vector and β ≥ 0.
One way to think about this instance is as each data point x consisting of a signal part
where g0 is a one-dimensional standard gaussian (a gaussian multiple of a fixed vector
noise part g ∼ N (0, I) (independent of g0. Then x = g +
βg0v is a gaussian random variable
√
√
√
βg... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
nice papers about this and similar phenomena, including [Pau, Joh01, BBAP05, Pau07,
BS05, Kar05, BGN11, BGN12].5
In what follows we will find the critical value of β and estimate the location of the largest eigenvalue
of Sn. While the argument we will use can be made precise (and is borrowed from [Pau]) we will
In short... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
1 ∈ R, denote, respectively, an eigenvalue and
associated eigenvector for Sn. By the definition of eigenvalue and eigenvector we have
√
1 (cid:20) (1 + β)ZT
1 + βZT
n
√
1 Z1
2 Z1
1 Z2
1 + βZT
ZT
2 Z2
(cid:21) (cid:20)
(cid:21)
v1
v2
ˆ= λ
(cid:21)
(cid:20) v1
v2
which can be rewritten as
1
n
(1 + β)ZT
1
1 Z1v1 + (cid:112... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
=
ˆ
λv1
If v1 = 0 (again, not properly justified here, see [Pau]) then this means that
ˆ
λ = (1 + β)ZT
1
n
1
1 Z1 + (cid:112)1 + βZT
n
1 Z2
(cid:18)
1
ˆ
λ I − ZT
n
2 Z2
(cid:19)
− 1
1
n
(cid:112)1 + βZT
2 Z1
(18)
First observation is that because Z1 ∈ Rn has standard gaussian entries then 1 ZT
n 1 Z1 → 1, meaning
that
(... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
−1 1
n
nU D V
1/2 T
(cid:17)
T
(cid:21)
Z1
T Z
T
(cid:1)
1 D1/
2
V T
(cid:16)
ˆ
λ I −V DV T
(cid:17)
−
1
(cid:0)
V D1/2 U T Z1
(cid:1)
(cid:21)
= (1 + β) +
T
(cid:1)
1
D1/2V T
(cid:16)
V
(cid:104)
ˆ
λ I −D
(cid:105)
V T (cid:17)
−
1
D1
(cid:0)
/2 U T
V
Z1
(cid:21)
(cid:1)
U T Z
= (1 + β)
(cid:20)
1 + (cid:0)U T Z (cid:... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
16)(cid:104)ˆ
1
n
p
−1
(cid:88)
j=1
g2 Djj
j ˆλ − Djj
= (1 + β)
1 +
Because we expect the diagonal entries of D to be distributed
according to the Marchenko-Pastur
distribution and g to be independent to it we expect that (again, not properly justified here, see [Pau])
p−1
1 (cid:88)
p − 1
j=1
g2 D j
j ˆ
j
λ − D... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
γ
β
> γ+.
Another important question is wether the leading eigenvector actually correlates with the planted
perturbation (in this case e1). Turns out that very similar techniques can answer this question as
well [Pau] and show that the leading eigenvector vmax of Sn will be non-trivially correlated with e1 if
and only ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
λmax
(cid:18) 1
n
√ W + ξvvT
(cid:19)
1
→ ξ + .
ξ
(21)
1.3.2 An open problem about spike models
Open Problem 1.3 (Spike Model for cut–SDP [MS15]. As since been solved [MS15]) Let
W denote a symmetric Wigner matrix with i.i.d. entries Wij ∼ N (0, 1). Also, given B ∈ Rn×n sym-
metric, define:
Define q(ξ) as
Q(B) = max {Tr(... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
:104)
(cid:16)
Since 1 E Tr 11T
n
These observ
ξ 11T + √1 W
n
imply that 1
n
(cid:17)(cid:105)
≈ ξ, by taking X = 11T we expect that q(ξ) ≥
ξ.
ations
≤ ξ < 2 (see [MS15]). A reasonable conjecture is that it is equal
to 1. This would imply that a certain semidefinite programming based algorithm for clustering under
the S... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
graph. The number of connected components is simply the size of the smallest partition of
the nodes into connected subgraphs. The Petersen graph is connected (and thus it has only 1
connected component).
• A clique of a graph G is a subset S of its nodes such that the subgraph corresponding to it is
complete. In other ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
.
r(G) := max {c(G), c (Gc)} .
Given r, let R(r) denote the smallest integer n such that every graph G on n nodes must have r(G) ≥ r.
Ramsey [Ram28] showed that R(r) is finite, for every r.
Remark 2.1 It is easy to show that R(3) ≤ 6, try it!
We will need a simple estimate for what follows (it is a very useful consequen... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
∈( r )
We will proceed by estimating E [X]. Note that, by linearity of expectation,
E [X] =
(cid:88) E [X(S)] ,
S∈(V
r )
and E [X(S)] = Prob {S is a clique or independent set} = 2
(|S
2 2 )
|
.
This means that
E [X] =
By Proposition 2.2 we have,
(cid:88) 2
r ) 2(|S|
2 )
∈(V
S
=
(cid:18)n
r
(cid:19) 2
2(r
2)
=
(cid:18)n... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
the definition of R(r) above, the following questions are open:
• What is the value of R(5)?
• What are the asymptotics
of R(s)? In particular, improve on the base of the exponent on either
the lower bound (
2) or the upper bound (4).
√
27 • Construct a family of graphs G = (V, E) with increasing number of vertices for ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
1 . Then, with high
| c
2
|
R(G) ≤ 2 log2(n).
Proof. Given n, we are interested in upper bounding Prob {R(G) ≥ (cid:100)2 log2 n(cid:101)}. and we proceed by
union bounding (and making use of Proposition 2.2):
Prob {R(G) ≥ (cid:100)2 log2 n(cid:101)} = Prob (cid:8)
(cid:91)
(cid:88)
{S is a clique or independe... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
EH89]) Prove or disprove the following:
For any finite graph H, there exists a constant δH > 0 such that any graph on n nodes that does
not contain H as a subgraph (is a H-free graph) must have
r(G) (cid:38) nδH .
(cid:50)
√
It is known that r(G) (cid:38) exp (cid:0)cH
log n , for some constant cH > 0 (see [Chu13] for a... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
> 2 log2 n this clique is larger
than any other clique that was in the graph before
planting. This means that, if ω > 2 log2 n, there is enough information in the graph to find the planted
clique. In fact, one can simply look at all subsets of size 2 log2 n + 1 and check wether it is clique: if
it is a clique then it ve... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
whether the planted clique contains them.
√
29 2. Is there a polynomial time algorithm that is able to distinguish, with high probability, G from a
draw of G (cid:0)n, 1
2
(cid:1) for ω (cid:28)
√
n. For example, for ω ≈
√
n
log n .
3. Is there a quasi-linear time algorithm able to find the largest clique of G (with hig... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
W.
If we start a random walker at node i (X(0) = 1) then the probability that, at step t, is at node j
is given by
ij
In other words, the probability cloud of the random walker at poin
t t, given that it started at node i
is given by the row vector
Prob {X(t) = j|X(0) = i} = (cid:0)M t
(cid:1)
.
Prob {X(t)|X(0) = i} = ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
�n].
1
M = ΦΛΨT ,
and Φ, Ψ form a biorthogonal system in the sense that ΦT Ψ = In n or, equivalently, ϕT
Note that ϕk and ψk are, respectively right and left eigenvectors of M , indeed, for all 1 ≤ k ≤ n:
j ψk = δ
×
jk.
Also, we can rewrite this decomposition as
M ϕk = λkϕk
and ψT
k M = λkψT
k .
and it is easy to see t... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
). Then,
λkϕk (imax) = M ϕk (imax) =
n
(cid:88)
j=1
Mimax,jϕk (j) .
This means, by triangular inequality that, that
|λk| =
n
(cid:88)
j=1
|
M max,j
i
|
k (j)
|
ϕ
|
ϕ (i
|
k max
≤
|
)
n
(cid:88)
j=1
M
|
i
max,j
| = 1.
(cid:50)
Remark 2.8 It is possible that there are other eigenvalues with magnitude 1 but only if G is d... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
This motivates truncating the
Diffusion Map by taking only the first d coefficients.
Definition 2.10 (Truncated Diffusion Map) Given a graph G = (V, E, W ) and dimension d, con-
struct M and its decomposition M = ΦΛΨT as described above. The Diffusion Map truncated to d
dimensions is a map φt : V → Rd given by
(d)φt
(vi) =
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
)
j=1
1
deg(j)
(cid:34) n
(cid:88)
k=1
λt
k (ϕk(i1) − ϕk(i2)) ψk(j)
(cid:35)2
n
(cid:88)
j=1
1
deg
(j)
(cid:34)
n
(cid:88)
k=1
t
λk (ϕk(i1) − ϕk(i2)) ψk(j)
(cid:35)2
=
=
n (cid:34) n
(cid:88)
(cid:88) t
λk (ϕk(i1)
− ϕk(i2))
(cid:35)
2
ψk(j)
deg(j)
(cid:112)
k=1
j=1
(cid:13)
n
(cid:13)
(cid:13)(cid:88)
(cid:13)
(cid:13)... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
of examples
The ring graph is a graph on n nodes {1, . . . , n} such that node k is connected to k − 1 and k + 1 and
1 is connected to n. Figure 2 has the Diffusion Map of it truncated to two dimensions
Another simple graph is Kn, the complete graph on n nodes (where every pair of nodes share an
edge), see Figure 3.
33 ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
goal is for the graph to capture the structure of the manifold. To each data point we will
associate a node. For this we should only connect points that are close in the manifold and not points
that maybe appear close in Euclidean space simply because of the curvature of the manifold. This
is achieved by picking a smal... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
if the square or stripe moves to the right all the way to the end of the screen, it shows
up on the left side (and same for up-down in the two-dimensional case). Not only this point cloud
should have a one dimensional structure but it should also exhibit a circular structure. Remarkably,
this structure is completely ap... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
,
let’s say that there are two labels, {−1, +1}.
Let’s say we are given the task of labeling point “?” in Figure 10 given the labeled points. The
natural label to give to the unlabeled point would be 1.
However, let’s say that we are given not just one unlabeled point, but many, as in Figure 11; then
it starts being ap... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
to node i.
We thus are interested in solving
min
f :V →R: f (i)=fi i=1,...,l
(cid:88)
i<j
wij (f (i) −
2
f (j)) .
If we denote by f the vector (in Rn with the function values) then we are can rewrite the problem
as
wij (f (i) − f (j)) =
2
(cid:88)
i<j
=
=
(cid:88)
i<j
(cid:88)
i<j
(cid:88)
i<j
wij [(ei − ej) f ] [(ei −... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
−
(cid:90)
f (x)f (cid:48)(cid:48)(x)dx.
(cid:90)
(cid:107)∇f (x)(cid:107)2
dx
=
(cid:90)
d
(cid:88)
k=1
(cid:18) ∂
f
∂xk
(cid:19)2
(x)
dx = B. T. −
(cid:90)
f (x)
d
(cid:88)
k=1
∂2f
∂x2
k
(x)dx = B. T. −
(cid:90)
f (x)∆f (x)dx,
which helps motivate the use of the term graph Laplacian.
Let us consider our problem
min
f... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
.
−
−1
Remark 2.13 The function f function constructed is called a harmonic extension. Indeed, it shares
properties with harmonic functions in euclidean space such as the mean value property and maximum
principles; if vi is an unlabeled point then
f
(cid:2)
(i) = D−1
u (Wulfl + Wuufu)(cid:3) =i
1
(i)
deg
n
(cid:88)
j=1... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
Let’s say that we wan
(cid:82)
sphere, that minimizes
centered at 0 with unit radius)
B0(1)
t to find a function in Rd that takes the value 1 at zero and −1 at the unit
(cid:107)∇f (x)(cid:107)2dx. Let us consider the following function on B0(1) (the ball
x
fε( ) =
|
(cid:26) 1 − 2 |x
−1
ε
if
|x| ≤ ε
otherwise.
A quick ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
not f T Lf but f T L2f instead,
Figure 15 shows the outcome of the same experiment with the f T Lf replaced by f T L2f and con-
firms our intuition that the discontinuity issue should disappear (see, e.g., [NSZ09] for more on this
phenomenon).
2
41 Figure 11: In this example we are given many unlabeled points, the unlab... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
[Llo82] (also known as the k-means algorithm), is an iterative algorithm that
alternates between
• Given centers µ1, . . . , µk, assign each point xi to the cluster
l = argminl=1,...,k (cid:107)xi − µl(cid:107) .
• Update the centers µl = 1
|Sl|
(cid:80)
i∈S
x
i.
l
Unfortunately, Lloyd’s algorithm
is not guaranteed to ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
always convex clusters. This means that k-means may have diffi-
culty in finding cluster such as in Figure 17.
3.2 Spectral Clustering
A natural way to try to overcome the issues of k-means depicted in Figure 17 is by using Diffusion
Maps: Given
the data points we construct a weighted graph G = (V, E, W ) using a kernel K(... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
(Spectral Clustering) Given a graph G = (V, E, W ) and a number of clusters k
(and t), Spectral Clustering consists in taking a (k − 1) dimensional Diffusion Map
(kφ −1)
t
(i) =
λt
2ϕ2(i)
.
..
λt
kϕk(i)
and clustering the points φt
tering.
(k 1)
− (1), φt
(k 1)
− (2), . . . , φt
(k 1)
− (n) ∈ Rk−1 using, for... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
, W ), a natural measure to measure a vertex partition (S, Sc) is
cut(S) =
(cid:88) (cid:88)
i∈S j∈Sc
wij.
Note however that the minimum cut is achieved for S = ∅ (since cut(∅) = 0) which is a rather
meaningless choice of partition.
Remark 3.3 One way to circumvent this issue is to ask that |S| = |Sc| (let’s say that t... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
�V
A similar object is the Normalized Cut, Ncut, which is given by
Ncut(S) =
cut(S)
vol(S)
+
cut(Sc)
vol(Sc)
.
Note that Ncut(S) and h(S) are tightly related, in fact it is easy to see that:
h(S) ≤ Ncut(S) ≤ 2h(S).
13W is the matrix of weights and D the degree matrix, a diagonal matrix with diagonal entries Dii = deg(i... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
1 = pT
pT
Proposition 3.5 Given a graph G = (V, E, W ) and a partition (S, Sc) of V , Ncut(S) corresponds
to the probability, in the random walk associated with G, that a random walker in the stationary
distribution goes to Sc conditioned on being in S plus the probability of going to S condition on being
in Sc, more e... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
balanced partition.
Recall that balanced partition can be written as
1
min
4 y∈{−1,1}n
1T y=0
yT LGy.
An intuitive way to relax the balanced condition is to allow the labels y to take values in two
different real values a and b (say yi = a if i ∈ S and yj = b if i ∈/ S) but not necessarily ±1. We can
then use the notion... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
=
(cid:16) vol(Sc)
vol(S) vol(G)
(cid:16)
−
vol(S)
vol(Sc) vol(G)
(cid:17) 1
2
(cid:17) 1
2
if i ∈ S
if i ∈ Sc.
49 wij(y
i
2
− yj)
w
2
ij(yi − yj)
Proof.
yT LGy =
=
=
=
=
=
1 (cid:88)
2
i,j
(cid:88) (cid:88)
i∈S j∈Sc
(cid:88) (cid:88)
i∈S j S
∈
c
(cid:88)
(cid:88)
i∈S j
∈S
c
(cid:88) (cid:88)
i∈S j∈S
c
(cid:88)... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
is, in general, NP-hard, we consider a similar problem where the constraint that
y can only take two values is removed:
min yT LGy
s. t. y ∈ Rn
yT
Dy = 1
yT D1 = 0.
(27)
Given a solution of (27) we can round it to a partition by setting a threshold τ and taking
S = {i ∈ V : yi ≤ τ }. We will see below that (27) is an e... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
relaxation (27) is obtained from (26) by removing a constraint we immediately have
that
This means that
λ2 (LG) ≤ min Ncut(S).
S⊂V
1
2
λ2 (LG) ≤ hG.
In what follows we will show a guarantee for Algorithm 3.2.
Lemma 3.8 There is a threshold τ producing a partition S such that
This implies in particular that
h(S) ≤ (cid:... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
y = ϕ2 satisfies the conditions and gives δ = λ2 (LG) this proves the Lemma.
We will pick this threshold at random and use the probabilistic method to show that at least one
of the thresholds works.
First we can, without loss of generality, assume that y1 ≤ · ≤ yn (we can simply relabel the
vertices). Also, note that sc... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
2
∈V
i∈V j
(cid:88)
(cid:88)
i∈V j
∈V
w
ij Prob{(S, Sc) cuts the edge (i, j)}
Note that Prob{(S, Sc) cuts the edge (i, j)} is
(cid:12)
(cid:12)x2
2
(cid:12)
i and xj have the same sign and xi +xj
otherwise. Both cases can be conveniently upper bounded by |xi − xj| (|xi| + |xj|). This means that
(cid:12)
(cid:12)
(cid:1... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
{xi is in the smallest set (in terms of volume)},
to break ties, if vol(S) = vol(S
i=1
c) we take the “smallest” set to be the one with the first indices.
Note that m is always in the largest set. Any vertex j < m is in the smallest set if xj ≤ τ ≤ xm = 0
and any j > m is in the smallest set if 0 = xm ≤ τ ≤ xj. This mea... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
cases where:
hG ≤ φ or hG ≥ 2
φ. Can this be improved?
√
Open Problem 3.1 Does there exists a constant c > 0 such that it is N P -hard to, given φ, and G
distinguis between the cases
1. hG ≤ φ, and
√
2. hG ≥ c
φ?
It turns out that this is a consequence [RST12] of an important conjecture in Theoretical Computer
Science ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
14
Ayt
(cid:0)
(cid:1)
One drawback of the power method is that when using it, one cannot be sure, a posteriori, that
there is no eigenvalue of A much larger than what we have found, since it could happen that all our
guesses were orthogonal to the corresponding eigenvector.
It simply guarantees us that if such an
eige... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
necessarily forming a partition).
Another natural definition is
ϕG(k) =
min
S:vol S≤ 1
k vol(G)
cut(S)
vol(S)
.
ϕG(k) ≤ ρG(k).
It is easy to see that
The following is known.
Theorem 3.11 ([LGT12]) Let G = (V, E, W ) be a graph and k a positive integer
Also,
ρG(k) ≤ O (cid:0)k2(cid:1) (cid:112)
λk,
ρG(k) ≤ O
(cid:16)(cid... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
many more concentration inequalities. Chebyshev’s
inequality is a simple inequality that control fluctuations from the mean.
Theorem 4.2 (Chebyshev’s inequality) Let X be a random variable with E[X 2] < ∞. Then,
Prob{|X − EX| > t} ≤
Var(X)
t2
.
Proof. Apply Markov’s inequality to the random variable (X − E[X])2 to get:
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
�
The inequality implies that fluctuations larger than O (
2n log n we get that the probability is at most 2 .n
for t = a
(cid:80)
n
Proof. We first get a probability bound for the event
i=1 Xi > t. The proof, again, will follow
from Markov. Since we want an exponentially small probability, we use a classical trick that ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
(λa)2/2.
Prob
(cid:40)
n
(cid:88)
i
=1
(cid:41)
Xi > t
≤ e−tλ
n
(cid:89)
i=1
2
e(λa) /2
15This follows immediately from the Taylor expansions: cosh(x) = (cid:80)∞
n=0 (2n)! , ex2/2 = (cid:80)∞
n=0
x2n
2n
n!
,
and (2n)!
≥ 2nn!.
= e−tλen(λa)2/2
nx
2
57 This inequality holds for any choice of λ ≥ 0, so we choose the value... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
�� 1 with probability p/2
0 with probability 1 − p
1 with probability p/2.
Then, E(ri) = 0 and |ri| ≤ 1 so Hoeffding’s inequality gives:
Prob
(cid:40)(cid:12)
n
(cid:12)
(cid:12)(cid:88)
(cid:12)
(cid:12)
i=1
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
ri
(cid:41)
> t
≤
2 exp
2
(cid:18) t
−
2n
(cid:19)
.
Intuitively,... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
:12)
(cid:12) Xi > t
(cid:12)
(cid:12)
(cid:12)
i=1
≤
2 exp
−
t2
2nσ2 + 2
3 at
.
Remark 4.6 Before proving Bernstein’s Inequality, note that on the example of Remark 4.4 we get
Prob
(cid:40)(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
n
(cid:88)
i=1
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
ri
(cid:41)
(cid:32)
> t
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
:16) λa − 1 − λa
e
(cid:17)
≤ 1 +
= 1 +
= 1 +
σ2
a2
Therefore,
Prob
(cid:40)
n
(cid:88)
i=1
(cid:41)
Xi > t
≤ e−λt
(cid:20)
1 +
(cid:16)
σ2
a2
eλa − 1 − λa
(cid:17)(cid:21)n
59 We will use a few simple inequalities (that can be easily proved with calculus) such as16 1 + x ≤
ex, for all x ∈ R.
This means that,
which rea... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
(cid:88)
Prob
(cid:41)
(cid:18)
Xi > t
≤ exp
−
2
(cid:19)
nσ
a2 {(1 + u) log(1 + u) − u}
i=1
The rest of the proof follo
ws by noting that, for every u > 0,
which implies:
(1 + u) log(1 + u) − u ≥
u
2 + 2
3
u
,
Prob
(cid:40)
n
(cid:88)
i=1
(cid:41)
(cid:32)
Xi > t
≤ exp −
(cid:32)
= exp
−
16In fact y = 1 + x is a tange... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
inside the exponent): Prob {|F (X) − EF (X)| ≥ t} ≤ 2 exp
. This exposition follows closely
σ2
the proof of Theorem 2.1.12 in [Tao12] and the original argument
is due to Maurey and Pisier. For
a proof with the optimal constants see, for example, Theorem 3.25 in these notes [vH14]. We will
also assume the function F is ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
= E [exp (λF (X))] E [exp (−λF (Y ))] ≥ E [exp (λF (X))]
Now we use the Fundamental Theorem of Calculus in a circular arc from X to Y :
F (X) − F (Y ) =
π
(cid:90) 2
0
∂
∂θ
F (Y cos θ + X sin θ) dθ.
61 The advantage of using the circular arc is that, for any θ, Xθ := Y cos θ + X sin θ is another random
(cid:48) = −Y si... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
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