text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
of Wo is easy via the Lyapunov equation
WoA + A�Wo = −C �C
SVD of a Hankel operator H can be expressed in terms of its Gramians:
Let wi be the normalized eigenvectors of R = Wc
WoWc
, i.e.
1/2
1/2
Rwi = �iwi, �1 → �2 → . . . , �m > 0, �m+1 = 0
The SVD of H is given by
H =
m
⎦
k=1
�
uk λk vk ,
1/2
where λk = ... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/2abdecbaa80cd2d06ffa931ca7ca34c3_lec8_6245_2004.pdf |
2.160 Identification, Estimation, and Learning
Lecture Notes No. 2
February 13, 2006
2. Parameter Estimation for Deterministic Systems
2.1 Least Squares Estimation
u
1
u
2
M
mu
Deterministic
System
w/parameter
θ
M
y
Linearly parameterized model
Input-output
y = u b 1 + u b 2 + K + b u
2
1
m m
Param... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/2ada973a6a0171832bafbb05c177fcc9_lecture_2.pdf |
)]1
(
t u b
2
t u − 2)
t y ) =
(
ϕ (t ) = [
− 1) +
(
(
(
(
Using an estimated parameter vector θ ˆ , we can write a predictor that predicts the output
from inputs:
T ˆ
ˆ(
t y θ ) = ϕ (t ) θ
(2)
1
We evaluate the predictor’s performance by the squared error given by
N1
VN (θ) = ∑(
N t =1
Problem: Find th... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/2ada973a6a0171832bafbb05c177fcc9_lecture_2.pdf |
��
where P = ∑ (ϕ(t)ϕ (t))
t =1
N
T
B = ∑ t y )ϕ(t)
(
t=1
−1
= (ΦΦ
T )
−1
(8)
(9)
(10)
2.2 The Recursive Least-Squares Algorithm
While the above algorithm is for batch processing of whole data, we often need to
estimate parameters in real-time where data are coming from a dynamical system.
2
A... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/2ada973a6a0171832bafbb05c177fcc9_lecture_2.pdf |
T )
3
B = ∑ t y )ϕ (t )
(
N
t = 1
Three steps for obtaining a recursive computation algorithm
a) Splitting Bt and Pt
From (10)
Bt = ∑ i y )ϕ (i ) = ∑ i y )ϕ (i ) +
(
(
t y )ϕ (t )
(
t − 1
t
i = 1
i = 1
Bt = Bt −1 +
t y )ϕ (t )
(
From (11)
t
− 1 = ∑ (ϕ (i )ϕ (i ))
P t
T
i = 1
P=
t − 1
−1
P
t
− ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/2ada973a6a0171832bafbb05c177fcc9_lecture_2.pdf |
t − 1
P ϕ (t )ϕ ( P
t
t
)
t − 1
T
T
Pt − 1ϕ (t )ϕ ( P
)
t
t − 1
= ( 1 +ϕ ( P
t ϕ (t ))
)
T
t − 1
(18)
P t − 1 − Pt
Therefore,
P t − 1ϕ (t )ϕ ( P
)
t
t − 1
Pt = Pt − 1 − ( 1 +ϕ ( P
t ϕ (t ))
)
Note that no matrix inversion is needed for updating Pt !
This is a special case of the Matrix Inversion L... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/2ada973a6a0171832bafbb05c177fcc9_lecture_2.pdf |
Pt − 1ϕ (t )
= ( 1 +ϕ ( P
T
) t − 1
ˆ
T
Replacing this by Κ t ∈ R m × 1 , we obtain (17)
The Recursive Least Squares (RLS) Algorithm
ˆ
Pt − 1ϕ (t )
ˆ
θ(t ) =θ (t − 1) + ( 1 +ϕ ( P
T
) t − 1
)
t
Pt − 1ϕ (t )ϕ ( P
t − 1
Pt = Pt − 1 − ( 1 +ϕ ( P
t ϕ (t ))
) t − 1
T
T
t ϕ (t ))(
t y ) −ϕ (t )θ (t − 1))
( ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/2ada973a6a0171832bafbb05c177fcc9_lecture_2.pdf |
3. Convex functions
Convex Optimization — Boyd & Vandenberghe
basic properties and examples
operations that preserve convexity
the conjugate function
quasiconvex functions
log-concave and log-convex functions
convexity with respect to generalized inequalities
•
•
•
•
•
•
3–1
Definition
f : Rn
→
R is conve... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
a, b
R
powers: xα on R++, for 0
α
≤
logarithm: log x on R++
∈
1
≤
•
•
•
•
•
•
•
•
Convex functions
3–3
Examples on Rn and Rm×n
affine functions are convex and concave; all norms are convex
examples on Rn
•
norms:
affine function f (x) = aT x + b
�p = (
x
•
�
examples on Rm×n (m n matrices)
p)1/p fo... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
R with f (X) = log det X, dom f = S++
n
g(t) = log det(X + tV ) =
log det X + log det(I + tX −1/2V X −1/2)
n
=
log det X +
log(1 + tλi)
�
i=1
where λi are the eigenvalues of X −1/2V X −1/2
g is concave in t (for any choice of X
0, V ); hence f is concave
≻
Convex functions
3–5
Extended-value extension
e... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
�x1
�
, . . . ,
∂f (x)
∂xn
�
exists at each x
dom f
∈
1st-order condition: differentiable f with convex domain is convex iff
f (y)
≥
f (x) +
∇
f (x)T (y
−
x)
for all x, y
dom f
∈
f (y)
f (x) + ∇f (x)T (y − x)
(x, f (x))
first-order approximation of f is global underestimator
Convex functions
3–7
Second-o... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
f (x) =
Ax
�
b
2
2
�
−
f (x) = 2AT (Ax
b),
−
∇
2f (x) = 2AT A
∇
convex (for any A)
quadratic-over-linear: f (x, y) = x2/y
2f (x, y) =
∇
2
3
y
�
y
x
−
y
x
−
� �
T
0
�
�
)
y
,
x
(
f
2
1
0
2
convex for y > 0
Convex functions
1
y
0
0 −2 x
2
3–9
log-sum-exp: f (x) = log
n
=1 exp xk is convex
k
... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
of f : Rn
R:
→
Cα =
x
{
∈
dom f
f (x)
α
}
≤
|
sublevel sets of convex functions are convex (converse is false)
epigraph of f : Rn
R:
→
epi f =
(x, t)
{
∈
Rn+1
x
|
∈
dom f, f (x)
t
}
≤
epi f
f
f is convex if and only if epi f is a convex set
Convex functions
3–11
Jensen’s inequality
basic inequality: if... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
•
•
Convex functions
3–13
Positive weighted sum & composition with affine function
nonnegative multiple: αf is convex if f is convex, α
0
≥
sum: f1 + f2 convex if f1, f2 convex (extends to infinite sums, integrals)
composition with affine function: f (Ax + b) is convex if f is convex
examples
log barrier for linea... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
is convex in x for each y
, then
∈ A
g(x) = sup f (x, y)
y∈A
is convex
examples
•
•
•
support function of a set C: SC(x) = supy∈C yT x is convex
distance to farthest point in a set C:
f (x) = sup
y∈C �
x
y
�
−
maximum eigenvalue of symmetric matrix: for X
Sn ,
∈
λmax(X) =
sup y T Xy
kyk2=1
Convex fun... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
if
gi convex, h convex, h˜ nondecreasing in each argument
gi concave, h convex, h˜ nonincreasing in each argument
proof (for n = 1, differentiable g, h)
f ′′ (x) = g ′ (x)T
2h(g(x))g ′ (x) +
h(g(x))T g ′′ (x)
∇
∇
examples
m
i=1
�
log
log gi(x) is concave if gi are concave and positive
m
i=1 exp gi(x) is con... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
/t),
dom g =
(x, t)
{
x/t
|
∈
dom f, t > 0
}
g is convex if f is convex
examples
f (x) = xT x is convex; hence g(x, t) = xT x/t is convex for t > 0
negative logarithm f (x) =
g(x, t) = t log t
−
t log x is convex on R2
++
log x is convex; hence relative entropy
−
if f is convex, then
g(x) = (c T x + d)f
� ... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
=
(1/2)x T Qx)
−
sup (y T x
x
1
2
y T Q−1 y
Convex functions
3–22
Quasiconvex functions
f : Rn
→
R is quasiconvex if dom f is convex and the sublevel sets
Sα =
x
{
∈
dom f
f (x)
α
}
≤
|
are convex for all α
β
α
a
b
c
f is quasiconcave if
f is quasiconvex
−
f is quasilinear if it is quasiconvex and q... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
(to us if xi > 0)
we assume x0 < 0 and x0 + x1 +
+ xn > 0
· · ·
present value of cash flow x, for interest rate r:
PV(x, r) =
n
�
i=0
(1 + r)−i xi
internal rate of return is smallest interest rate for which PV(x, r) = 0:
•
•
•
•
IRR(x) = inf
r
{
0
PV(x, r) = 0
}
|
≥
IRR is quasiconcave: superlevel set ... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
)y)
−
≥
f (x)θf (y)1−θ
for 0
θ
≤
≤
1
f is log-convex if log f is convex
•
•
•
powers: xa on R++ is log-convex for a
≥
many common probability densities are log-concave, e.g., normal:
≤
0, log-concave for a
0
f (x) =
1
(2π)n det Σ
−1(x−x¯)T Σ−1(x−x¯)
e
2
�
cumulative Gaussian distribution function Φ is ... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
C
⊆
•
Rn convex and y is a random variable with log-concave pdf then
f (x) = prob(x + y
C)
∈
is log-concave
proof: write f (x) as integral of product of log-concave functions
f (x) =
�
g(x + y)p(y) dy,
g(u) =
1 u
0 u
C
C,
∈
�∈
�
p is pdf of y
Convex functions
3–29
example: yield function
Y (x) = prob... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
X, i.e.,
�2
�
z T (θX + (1
for X, Y
Sm , 0
θ
≤
≤
∈
−
1
θ)Y )2 z
≤
θzT X 2 z + (1
θ)z T Y 2 z
−
therefore (θX + (1
θ)Y )2
−
�
θX 2 + (1
θ)Y 2
−
Convex functions
3–31
MIT OpenCourseWare
http://ocw.mit.edu
6.079 / 6.975 Introduction to Convex Optimization
Fall 2009
For information about citing these ma... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/2ae23d35685ff402473b36011138149a_MIT6_079F09_lec03.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.685 Electric Machines
Class Notes 6: DC (Commutator) and Permanent Magnet Machines
(cid:13)c 2005 James L. Kirtley Jr.
1
Introduction
Virtually all electric machines, and all practical electric machines employ some form of... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
utator system. The interaction magnetic field is provided
(in this picture) by a field winding. A permanent magnet field is applicable here, and we will have
quite a lot more to say about such arrangements below.
Now, if we assume that the interaction magnetic flux density averages Br, and if there are Ca
conductors undern... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
of noting the apparent electric
field within a moving object (as in the conductors in a DC machine):
~E′
~
= E + ~v × B
~
Now, note that the armature conductors are moving through the magnetic field produced by
the stator (field) poles, and we can ascribe to them an axially directed electric field:
Ez = −RΩBr
2
rB
dlr
rv
... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
The two
“break points” are at zero speed and at the “zero torque” speed:
Ω0 =
Va
GIf
3
Ra
+
Va
-
+
Ω
G I f
-
Figure 3: DC Machine Equivalent Circuit
Electrical
Mechanical
Figure 4: DC Machine Operating Regimes
For 0 < Ω < Ω0, the machine is a motor: electric power in and mechanical power out are both
positive. For ... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
: Two-Chopper, separately excited machine hookup
+
V
-
Figure 6: Series Connection
not yield any meaningful ability to control speed and the simple applications to which it used to
be used are handled by induction machines.
Another connection which is still widely used is the series connection, in which the field win... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
RPM limit of AC motors, and this is one reason why they are so widely used:
with the high rotational speeds it is possible to produce more power per unit mass (and more power
per dollar).
1.3 Commutator:
The commutator is what makes this machine work. The brush and commutator system of this
class of motor involves quit... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
: Commutation Interpoles
In larger machines the commutation process would involve too much sparking, which causes
brush wear, noxious gases (ozone) that promote corrosion, etc. In these cases it is common to use
separate commutation interpoles. These are separate, usually narrow or seemingly vestigal pole
pieces which ... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
the same number of ampere-turns as
the armature. Normally they have the same number of turns and are connected directly in series
with the armature brushes. What they do is to almost exactly cancel the flux produced by the
armature coils, leaving only the main flux produced by the field winding. One might think of these
c... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
300
400
-0.4
-0.6
-0.8
-1
Kilo Am peres/Meter
a
l
s
e
T
Figure 11: Hysteresis Loop Of Ceramic Permanent Magnet
Permanent magnet materials are, at core, just materials with very wide hysteresis loops. Fig-
ure 11 is an example of something close to one of the more popular ceramic magnet materials. Note
that this hystere... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
is 10−4 Tesla. And, finally, an Oersted is that field
9
Demagnetization Curve
Energy Product Loci
a
l
s
e
T
,
B
Hc
Br
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-250
-200
-150
-100
-50
0
H, kA/m
Figure 12: Demagnetization Curve
intensity required to produce one Gauss in the permeability of free space. Since the perm... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
down in the air-gap.
Thus we are following the same reference direction as we go around the Ampere’s Law loop.
That becomes:
~H · d ℓ = Hmhm + Hgg
~
I
Now, Gauss’ law could be written for either the upper or lower piece of the magnetic circuit.
Assuming that the only substantive flux leaving or entering the magnetic cir... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
− ℘µ
u
0
Figure 15: Load Line, Unit Permeance Analysis
Permanent Magnet
Magnetic Circuit, µ→∞
Figure 16: Surface Magnet Primitive Problem
In the region of the magnet and the air-gap, Ampere’s Law and Gauss’ law can be written:
∇ · µ0
(cid:16)
~
Hm + M0
~∇ × H = 0
= 0
(cid:17)
∇ · µ0 ~Hg = 0
~
Now, if in the magnet the ... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
be made. We would produce the same air-gap flux density
if we regard the permanent magnet as having a surface current around the periphery equal to the
magnetization intensity. That is, if the surface current runs around the magnet:
This would produce an MMF in the gap of:
Kφ = M0
F = Kφhm
and then since the magnetic fie... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
1.1.
2. The “reluctance factor” ff is cited as being about 1.2.
We may further estimate the ratio of areas of the gap and magnet by:
Ag
A
m
=
R + g
2
R + g + h
m
2
Now, there are a bunch of approximations and hand wavings in this expression, but it seems to
work, at least for the kind of machines contemplated.
A second... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
coil-side waveforms that add with
a slight phase shift.
vc
0m
0 m
Figure 19: Voltage Induced in a Coil
If, on the other hand, the coil thrown is smaller than the magnet angle, the picture is the same,
only the width of the pulses is that of the coil rather than the magnet. In either case the average
voltage generated b... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/2aec052d0c82b73a7b6c72d78f3a796d_MIT6_685F13_chapter6.pdf |
Algorithmic Aspects of Machine Learning
Ankur Moitra
c© Draft date March 30, 2014
Algorithmic Aspects of Machine Learning
©2015 by Ankur Moitra.
Note: These are unpolished, incomplete course notes.
Developed for educational use at MIT and for publication through MIT OpenCourseware.
Contents
Contents
Preface
1 I... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
. . . . . . . . . . . . . . . . 50
4 Sparse Recovery
53
4.1 Basics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Uniqueness and Uncertainty Principles . . . . . . . . . . . . . . . . . 56
4.3 Pursuit Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
i
ii
CONT... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
. . . . . . . . . . . . . . . . . . . 89
6.4 Clustering-Free Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 91
6.5 A Univariate Algorithm
. . . . . . . . . . . . . . . . . . . . . . . . . 96
6.6 A View from Algebraic Geometry . . . . . . . . . . . . . . . . . . . . 101
7 Matrix Completion
105
7.1 Bac... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
This course will be organized around algorithmic issues that arise in machine learn
ing. The usual paradigm for algorithm design is to give an algorithm that succeeds on
all possible inputs, but the difficulty is that almost all of the optimization problems
that arise in modern machine learning are computationally int... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
matrix factorization
(b) topic modeling
(c) tensor decompositions
(d) sparse recovery
(e) dictionary learning
(f) learning mixtures models
(g) matrix completion
Hopefully more sections will be added to this course over time, since there are a
vast number of topics at the intersection of algorithms and machine l... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
read-off the best low
rank approximation to M from it.
Definition 2.1.1 The Frobenius norm of a matrix M is IM IF =
ternately, if M = r
T , IM IF =
i=1 uiσivi
2 .
σi
M 2 Al-
i,j.
i,j
Consider the following optimization problem: Let B be the best rank k ap
proximation to M in the Frobenius norm - i.e. B is the mini... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
to M in the operator norm is also attained by B =
k
i=1 uiσivi , in which case the error is IM − BI2 = σk+1.
T
Let us give one more interpretation of the singular value decomposition. We
can regard an m × n matrix M as a collection of n data points in Rm . We associate
a distribution Δ with this set of points whic... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
to reduce M
to bidiagonal form using Householder reflections, and then to compute the singular
value decomposition from this representation using the QR algorithm. Next we will
describe an application to text analysis.
Applications to Text Analysis
Latent Semantic Indexing: [49]
Suppose we are give a large collect... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
compute a
“similarity” score for document i and document j. We could do this by computing
�Mi, Mj �
where Mi is the ith column of M , etc. This function “counts” the number of words
in common. In particular, given a query we would judge how similar a document is
to it just be counting how many of its words occur i... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
sum to one. It is
not hard to see that if D is a diagonal matrix where the ith entry is the reciprocal
2.1. INTRODUCTION
9
of the sum of the entries in the ith column of A then M = AA WW where AA = AD and
WW = D−1W normalizes the data so that the columns of AA and of WW each sum to one.
Hence we are finding a set... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
identifying interesting instances of the problem.
The goal of this course is to not give up when faced with intractability, and to
look for new explanations. These explanations could be new models (that avoid the
aspects of the problem that allow us to embed hard problems) or could be identifying
conditions under w... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
is much smaller than m or n and with this in mind
we will instead ask: What is the complexity of this problem as a function of k?
We will make use of tools from algebra to give a polynomial time algorithm for any
k = O(1). In fact, the algorithm we present here will be nearly optimal in terms of
its dependence on k... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
. It is easy to see that the
columns of M are spanned by
⎫⎤⎡⎤⎡⎤⎡⎧
12
1
1
⎪⎪⎪⎬
⎪⎪⎪⎨
22
2
1
⎥
⎢
⎥
⎥
⎢
⎢
⎥
⎢
⎥
⎥
⎢
⎢
,
,
. . .
. . .
. .
⎥
⎢
⎥
⎥
⎢
⎢
⎦
⎣
⎦
⎦
⎣
⎣
.
⎪⎪⎪⎭
⎪⎪⎪⎩
2
n
n
1
It is easy to see that rank(M ) = 3 However, M has zeros along the diagonal and
non-zeros off it. Furthermore for any... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
1)
⎧
⎪⎨
⎪⎩
M = AW
A ≥ 0
W ≥ 0
This system consists of quadratic equality constraints (one for each entry of M ),
and linear constraints that A and W be entry-wise nonnegative. Before trying to
design better algorithms for k = O(1), we should ask a more basic question (whose
answer is not at all obvious):
Que... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
decompositions. This line of work culminated in algorithms whose running time is
exponential in the number of variables but is polynomial in all the other parameters
of the problem (the number of polynomial inequalities, the maximum degree and
the bit complexity of the coefficients). The running time is (nD)O(r) where... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
ic if there exist multivariate polynomials
p1, ..., pn such that
S = {x1, ..., xr|pi(x1, ..., xr) ≥ 0}
or if S is a finite union or intersection of such sets.
Definition 2.2.4 The projection of a semialgebraic set S is defined as
projS (X1, ..., X�) = {x1, ..., x�|∃ x�+1, ..., xr such that p(x1, ..., xr) ∈ S}
2.2. ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
1, ..., xr|B(p1(x1, ..., xr), ..., pn(x1, ..., xr)) = true}
is non-empty, and we assume that we can evaluate B (but not, say, that it has a
succinct circuit). A related result is the famous Milnor-Warren bound (see e.g. [7]):
Theorem 2.2.6 (Milnor-Warren) Given n polynomials p1, ..., pm of degree ≤ D
on r variables... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
4 Can we find an alternate system of polynomial inequalities that ex
presses the same decision problem but uses many fewer variables?
We will focus on a special case called simplicial factorization where rank(M ) = k.
In this case, we are asking whether or not rank+(M ) = rank(M ) = k and this
simplifies matters beca... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
have made progress since this system also has
nk + mk variables corresponding to the entries of A+ and W + . However consider
the matrix A+M . If we represent A+ as an k × n matrix then we are describing
its action on all vectors, but the crucial observation is that we only need to know
how A+ acts on the columns o... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
nm)O(k2) and solves the simplicial factorization problem.
The above approach is based on the paper of Arora et al [13] where the authors
also give a variable reduction procedure for nonnegative matrix factorization (in the
general case where A and W need not have full column or row rank respectively).
The authors r... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
give an algorithm
that runs in polynomial time (even for large values of r). Our discussion will revolve
around the intermediate simplex problem.
16
CHAPTER 2. NONNEGATIVE MATRIX FACTORIZATION
Intermediate Simplex Problem
Let us define the intermediate simplex problem:
We are given two polytopes Q and P with... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
(P0), intermediate simplex and the simplicial
factorization problem are each polynomial time interreducible. It is easy to see
that (P0) and the simplicial factorization problem are equivalent since in any two
factorizations M = U V or M = AW (where the inner-dimension equals the rank of
M ), the column spaces of M... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
xi. The description
of the complete reduction, and the proof of its soundness are involved (see [115]).
The trouble is that gadgets like those in Figure ?? are unstable. We can change
the number of solutions by small perturbations to the problem. Motivated by issues of
uniqueness and robustness, Donoho and Stodden ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
401k could be an anchor word for the topic
personal finance.
Why do anchor words help? It is easy to see that if A is separable, then the
rows of W appear as rows of M (after scaling). Hence we just need to determine
which rows of M correspond to anchor words. We know from our discussion in
Section 2.3 that (if we ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
start. Hence the anchor words
that are deleted are redundant and we could just as well do without them.
Separable NMF [13]
Input: matrix M ∈ Rn×m satisfying the conditions in Theorem 2.3.2
Output: A, W
Run Find Anchors on M , let W be the output
Solve for nonnegative A that minimizes IM − AW IF (convex programmin... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
will consider a related problem called topic modeling; see [28] for a compre
hensive introduction. This problem is intimately related to nonnegative matrix fac
torization, with two crucial differences. Again there is some factorization M = AW
but now we do not get access to M but rather WM which is a very crude appro... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
be
sparse while the latter is dense. Are there provable algorithms for topic modeling?
WM .
The Gram Matrix
We will follow an approach of Arora, Ge and Moitra [14]. At first this seems like a
fundamentally different problem than the ones we have considered because in this
model we cannot ask for longer documents, w... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
] as:
r
P[w1 = a, w2 = b|t1 = i, t2 = j]P[t1 = i, t2 = j]
i,j
and the lemma is now immediate. •
The key observation is that G has a separable nonnegative matrix factorization
given by A and RAT since A is separable and the latter matrix is nonnegative. Indeed
if RAT has full row rank then the algorithm in Theore... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
2 = w) =
P(word1 = w ' |word2 = anchor(t))P(t2 = t|w2 = w)
t
which we can think of a linear systems in the variables {P(t2 = t|w2 = w)}. It is
not hard to see that if R has full rank then it has a unique solution. Finally, we
compute the probabilities we were originally interested in by Bayes’ rule:
P(word w|topi... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
Run Find Anchors
Solve for P(topic t|word w) and use Bayes’ rule to compute A
End
Experiments
We are faced with a basic scientific question now: Are there really anchor words?
The following experiment was conducted in [12]:
2.4. TOPIC MODELS
23
(a) Run MALLET (a popular topic modeling toolkit) on a collection... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
regard T as a collection of p matrices
of size m × n that are stacked on top of each other.
We can generalize many of the standard definitions from linear algebra to the
tensor setting, however we caution the reader that while these parameters are easy
to compute for matrices, most parameters of a tensor are hard to... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
unit vectors) u1, u2
that
∈ R1000
M ≈ u1v T + u2v T
2
1
This is called factor analysis, and his results somewhat confirmed his hypothesis.
But there is a fundamental obstacle to this type of approach that is often referred
to as the “Rotation Problem”. Set U = [u1, u2] and V = [v1, v2] and let O be an
orthogonal ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
the same M ).
However if we are given a tensor
r
r
xi ⊗ yi ⊗ wi
T =
i=1
then there are general conditions (namely if {xi}i, {yi}i and {wi}i are each linearly
independent) not only is the true factorization the unique factorization of T with
rank r but in fact there are simple algorithms to find it! This is preci... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
imations do not necessarily share any
common rank one factors. In fact, subtracting the best rank one approximation to
a tensor T from it can actually increase its rank.
(c) For a real-valued matrix its rank over R and over C are the same, but this is
false for tensors.
There are real-valued tensors whose minimum ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
imation of T .
(cid:0)
⊗
1
1/n
1
1/n
1
1/n
(cid:0)
(cid:1)
(cid:1)
One last issue is that it is easy to see that a random n × n × n tensor will have
rank Ω(n2), but it is unknown how to explicitly construct any order three tensor
whose rank is Ω(n1+ε). And any such construction would give the first super-linear
... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
so many times) to phylogenetic reconstruction
[96], topic modeling [8] and community detection [9]. This decomposition also plays
a crucial role in learning mixtures of spherical Gaussians [75] and independent com
ponent analysis [36], although we will instead present a local search algorithm for
the latter problem... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
of Ta(Tb)+ and Tb(Ta)+ are U and V respectively
(after rescaling)
Proof: We can use the above formula for Ta and Tb and compute
Ta(Tb)+ = U DaDb
+U +
D+
Then almost surely over the choice of a and b we have that the diagonals of Da
b
will be distinct – this is where we use the condition that each pair of vectors ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
⊗ vi
i
That is the Khatri-Rao product of U and V of size m × r and n × r is an mn × r
matrix whose ith column is the tensor product of the ith column of U and the ith
column of V . The following lemma we leave as an exercise to the reader:
Lemma 3.1.7 If U and V are size m × r and n × r and have full column rank a... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
close to the true factors. In later sections, we will simply assume we are given
the true tensor T and what we present here is what justifies this simplification.
This section is somewhat technical, and the reader can feel free to skip it.
Recall that the main step in Theorem 3.1.3 is to compute an eigendecompo
sitio... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
x?
We have xA = M −1Ab = x + M −1e = x + M −1(Ab − b). So
Ix − xAI ≤
1
σmin(M )
Ib − AbI.
Since M x = b, we also have IbI ≤ σmax(M )IxI. It follows that
Ix − xAI
IxI
≤
σmax(M ) Ib − AbI
σmin(M ) IbI
= κ(M )
Ib − AbI
.
IbI
In other words, the condition number controls the relative error when solving a line... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
the previous section: is
M diagonalizable? Consider
W
U −1 WM U = D + U −1EU.
The proof that
WM is diagonalizable proceeds as follows:
Part (a) Since
M and U −1 WM U are similar matrices, they have the same set
W
of eigenvalues.
Part (b) Moreover we can apply Theorem 3.2.2 to U −1M U W = D + U −1EU
and if U is ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
eigenvalues of M have sufficient separation. It remains to check that
uAi ≈ ui. Let
=
r
cj uj = uAi.
j
Left-multiplying both sides of the equation above by W
M ,
Recall that W = M + E.
M
we get
r
cj λj uj + EuAi = λAiuAi. =⇒
r
cj (λj − λAi)uj = −EuAi.
j
j
T be the jth row of U −1 . Left-multiplying both side... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
and T represents
its true moments. Also TAa = TA(∗, ∗, a) → Ta and similarly for b.
We leave it as an exercise to the reader to check that TAb
+ under natural
T +
→ Ta b . We have already established that if
conditions. It follows that
E → 0, then the eigendecompositions of M and M + E converge. Finally we
concl... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
Open Question 1 Is there an efficient algorithm for tensor decompositions under
any natural conditions, for r = (1 + ε)n for any ε > 0?
For example, it is natural to consider a smoothed analysis model for tensor decompo
sition [26] where the factors of T are perturbed and hence not adversarially chosen.
The above uni... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
u. Alternatively, we can think of s(·) : V → Σ as a random function that assigns
states to vertices where the marginal distribution on s(r) is πr and
P uv = P(s(v) = j|s(u) = i),
ij
Note that s(v) is independent of s(t) conditioned on s(u) whenever the (unique)
shortest path from v to t in the tree passes through u... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
, Steel, Szekely, and
Warnow [57]. And from this, we can apply tensor methods to find the transition
matrices following the approach of Chang [36] and later Mossel and Roch [96].
Finding the Topology
The basic idea here is to define an appropriate distance function [109] on the edges
of the tree, so that we can appr... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
the topology. Fix four leaves a, b, c, and d, and
there are exactly three possible induced topologies between these leaves, given in
Figure 3.1. (Here by induced topology, we mean delete edges not on any shortest
path between any pair of the four leaves, and contract paths to a single edge if possi
ble). Our goal i... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
Handling Noise
Note that we can only approximate F ab from our samples. This translates into a
good approximation of ψab when a and b are close, but is noisy when a and b are
far away. The approach in [57] of Erdos, Steel, Szekely, and Warnow is to only use
quartets where all of the distances are short.
Finding th... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
also called Chang’s lemma
[36]).
In [96], Mossel and Roch use this approach to find the transition matrices of
a phylogenetic tree, given the tree topology, as follows. Let us assume that u and
v are internal nodes and that w is a leaf. Furthermore suppose that v lies on the
shortest path between u and w. The basic... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
the above algorithm, we need that the
transition matrices and the observation matrices are full-rank.
More precisely, we require that the transition matrices are invertible and that
the observation matrices whose row space correspond to a hidden node and whose
column space correspond to the output symbols each have... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
value bj = χS (Xj ) with
probably 2/3 and otherwise bj = 1 − χS (Xj ). The challenge is that we do not know
which labels have been flipped.
Claim 3.3.2 There is a brute-force algorithm that solves the noisy parity problem
using O(n log n) samples
Proof: For each T , calculate χT (Xj )bj over the samples. Indeed χT ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
the state of the ith internal node where si = χSi (X).
We can define the following transition matrices:
if i + 1 ∈ S
P i+1,i
=
if i + 1 ∈/ S
P i+1,i
=
(0, si)
(1, si + 1 mod 2)
⎧
1
⎪⎨
2
1
2
⎪⎩
0 otherwise
⎧
1
(0, si)
⎪⎨
2
1
.
(1, si)
2
⎪⎩
0 otherwise
At each internal node we observe xi and at th... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
each of which would lead to a different optimization problem e.g. sparsest
cut or k-densest subgaph.
However each of these optimization problems is N P -hard, and even worse are
hard to approximate. Instead, we will formulate our problem in an average-case
model where there is an underlying community structure that ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
graph G with high probability. If q − p is smaller, then it is not even
information theoretically possible to find π. Indeed, we should also require that
each part of the partition is large, and for simplicity we will assume that k = O(1)
and |{u|π(u) = i}| = Ω(n).
There has been a long line of work on partitioning ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
according to πu and πv respectively, and if they choose
the same community there is an edge with probability q and otherwise there is an
edge with probability p.
i=j πuπj
i
vq +
πuπi
i
(cid:80)
i
(cid:80)
Recall that in order to apply tensor decomposition methods what we really
need are conditionally independent... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
, b ∈ B and c ∈ C; then Ta,b,c is exactly the probability that
a random node x ∈ X is connected to a, b and c.
#
(cid:54)
(cid:54)
3.4. COMMUNITY DETECTION
43
This is immediate from the definitions above. In particular if we look at whether
(x, a), (x, b) and (x, c) are edges in G, these are conditionally in... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
in A, B and C and hence the factors {(ΠR)A}i,
{(ΠR)B }i, and {(ΠR)B }i will be non-negligibly far from linearly dependent.
i
i
i
Part (c) Note that if we have a good approximation to {(ΠR)A}i then we can
partition A into communities. In turn, if A is large enough then we can extend this
partitioning to the whole g... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
pure topic models (see
[10]). Recall that there is an unknown topic matrix A and we obtain samples from
the following model:
(a) Choose topic i for document j with probability pi
(b) Choose Nj words according to the distribution Ai
If each document has at least three words, we can define the tensor TA where TAa,b,c... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
Finally, in our application to phylogenetic reconstruction, each hidden node was in
one and only one state. Note however that in the context of topic models, it is much
more realistic to assume that each document is itself a mixture of topics and we will
refer to these as mixed models.
Latent Dirichlet Allocation
... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
bj ⊗ ck
T =
i,j,k
where D is r1 × r2 × r3. We call D the core tensor.
46
CHAPTER 3. TENSOR METHODS
This is different than the standard definition for a tensor decomposition where we
only summed over i = j = k. The good news is that computing a Tucker decom
position of a tensor is easy. Indeed we can always set... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
ability that the first three words in a random document are a, b and c respectively.
But we could just as well consider alternative experiments. The three experiments
we will need in order to given a tensor spectral algorithm for LDA are:
(a) Choose three documents at random, and look at the first word of each docu... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
=
(cid:80)
i,j,k Di,j,kAi ⊗ Aj ⊗ Ak
Proof: Let w1 denote the first word and let t1 denote the topic of w1 (and similarly
for the other words). We can expand P[w1 = a, w2 = b, w3 = c] as:
r
P[w1 = a, w2 = b, w3 = c|t1 = i, t2 = j, t3 = k]P[t1 = i, t2 = j, t3 = k]
i,j,k
and the lemma is now immediate. •
(cid:54)=
... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
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