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Lecture 21
SVD and Latent Semantic Indexing
and Dimensional Reduction
ShangHua Teng
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View Full Document Singular Value Decomposition
r
T
r
r
r
T
T
v
u
v
u
v
u
A
σ
2
1
2
2
2
1
1
1
≥
≥
≥
+
+
+
=
where
•
u
1
…
u
r
are the r orthonormal vectors that are basis of C(A) and
•
v
1
…
v
r
are the r orthonormal vectors that are basis of C(A
T
)
Low Rank Approximation
and Reduction
T
r
T
k
k
k
T
T
k
v
u
A
v
u
v
u
v
u
A
1
1
1
1
2
1
2
2
2
1
1
1
σ
=
≥
≥
≥
+
+
+
=
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View Full Document The Singular Value Decomposition
·
·
A
U
V
T
=
Σ
0
0
A
U
V
T
m x n
m x r
r x r
r x n
=
Σ
0
0
m x n
m x m
m x n
n x n
The Singular Value Reduction
·
·
A
U
V
T
m x n
m x r
r x r
r x n
=
Σ
0
0
A
k
U
k
V
k
T
m x n
m x k
k x k
k x n
=
Σ
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View Full Document How Much Information Lost?
T
r
T
k
k
k
T
T
k
v
u
A
v
u
v
u
v
u
A
1
1
1
1
2
1
2
2
2
1
1
1
σ
=
≥
≥
≥
+
+
+
=
Distance between Two Matrices
•
Frobenius Norm of a matrix A.
•
Distance between two matrices A and B
( 29
2
,
2
1
2
∑
∑∑

=

=
=
j
i
ij
ij
F
m
i
n
j
ij
F
B
A
B
A
A
A
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View Full DocumentHow Much Information Lost?
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This note was uploaded on 03/08/2010 for the course CS 232 taught by Professor Bera during the Spring '09 term at BU.
 Spring '09
 BERA
 Algorithms

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