Sobre el uso de la descomposición en valores singulares para

 1
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-$ A ' m × n & r ≤ min{m, n} % . r . 2r r / & r 0
" A$ & (m + n)r 1 2
$ 0
Ck ' (m + n)k$ k < r$ 3% $ . 0
4 . !$ !" 5& 0
0
Ck
A$ 0
4 Ck $ 0
' 0
A % $ 0
$ % . 6 4 6$ 3
0
. 0
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5$ 78$ $ " 9!"
W(Ck ) . 6 CK $ Ck /
3 W(Ck ) 2 . .$ 3$ " 4 $ %
.$ / k σi
A
Ck Ck . 6 - 3$ Ck & k 3
k $ ' m × n & r ≤ k 4$
σi A
Ak 7 % $ 512 × 512 3
: ;!
A$ ' m × n$ < .
=
A = USV T
"
U ' m × m$ V ' n × n & S m × n$ $ $ 0
σi $ i = 1, 2, · · · , min{m, n}$
0
A
σi = σ1 ≥ σ2 ≥ . . . 4 %
{σi } A ui
U vi V " 0
/" A - i$ (ui , σi , vi ) i A
4 > A
? $ 0
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A=
r
σi ui viT .
("
i=1
(ui , σi , vi ) i A 2" $ A 7 σi ui viT ;!=
Ak =
k
σi ui viT ,
k < r.
9"
i=1
0
Ak k
4 < A - Ak < A
!
A = (aij ) & B = (bij ) ' m × n$
> =
m,n
i,j=1 (aij − bij )
m,n 2
i,j=1 aij
D(A, B) =
2
;"
7 $ D(A, B) ≤ 0.05$ B < " A 4 . ! 0
Ak < A$ =
r
σi2
i=k+1
r
2
i=1 σi
D(A, Ak ) =
,"
- $ D(A, Ak ) 0
A
' k(m + n)=
Ck
A ∈ Rm×n $ r$ k < r 3 

uT1 v1T
 uT vT 
2 
 2
Ck =  
 
uTk vkT
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(u1 , . . . , uk ) k U &
(v1 , . . . , vk ) k V m & n > 0
1 ≤ m < n 4=
m
(m + n) < m n
"
2
= 1 1 ≤ m < n$ < > > p 1
1
> 2n
&
0
m + p = n$ m + n = n − p + n = 2n − p < 2n$ m+n
mn 0
mn
m
mn
>
=
m+n
2n
2
*"
A ' m × n 3 ≤ m ≤ n r ≥ m/2
< Ck )" 0
' 0
A
= @ m
−1
m 2
k=
+"
m−1
m 2
0
=
k<
m
2
"
& r ≥ m/2 k < r & > 3 2$ ' Ck k(m + n) A
m < n$ 0
" & 9$ =
m
(m + n)k < (m + n) < mn
"
2
0
/ m = n
4 ' Ck / k(m + n) = k2n < n2 (2n) = n2 $ 0
' A$ 0
4 ( < $ $ Ck r ≥ m/2
1/ $ & & Ck ' 0
A 4 $ =
/ ( & k ≥ 1$ +"$ < i = 1, . . . , k Ci ' 0
A
B 0
A ' 0
m > n$ ( AT - $ 0
9 3 9 " A ( r ≥ m/2
4 3
$ $ $ " 0
3
! " #
C '
512 × 512 512
498 × 621 498
256 × 256 239
359 × 657 359
$ $% & A 0
3 / ( W(A) . A$ ' TA $ ' 0
A$ & 3 %
3 W(A)
0
A = W −1 (TA ) < A 7 A A & TA 0
0
. " 4 $ % . W =
A
( 3 ( & ($ < Ck '
0
A - & ," & ( 3 k < r
9 . W Ck $ W(Ck ) TCk
. 3 W(Ck ) 3
TCk 0
%
k σi ; - $ . k 3
k &$ 0
C
W −1 (TCk ) = C
($ 9" AK 0
< A
4 % / W A & Ck 1
/ TA & TCK $ CK ' 0
A$ @$ . W(Ck ) Ck $ & k 0
Ak 3 A$ . < . 6 ( 3 .
" 6
& ! - 3% .
6 ( D . A ' m × n =
A −→ W(A) =
h1 d1
a1 v 1
("
h1 $ d1 $ a1 $ & v 1 ' m/2×n/2 & 8
/$ 8
$ & 8
$ a1 $ & '$ αi $ i = 1, . . . , m × n W(A)$ 0
=
A=
m×n
αi Wi = H 1 + D 1 + A1 + V 1
9"
i=1
Wi $ 0
.
6 & $ . Rm×n A1 < A 7 <
$ =
A1 = W −1
O O
a1 O
≡ W −1 (TA ) ≡ A
;"
. 6$ > 3 ' a1 $ m4n 1$ TA 3 a1 $v 1 & h1 $ > 3
A A1 $ V 1 & H 1 3m4 n / W a1 & .
6 ($ > 3 5 A = (aij ) 0
4
512 × 512$ & > 0
$ $ aij ∈ [0, 255] A r = 507 &$ $ (
5 k = 80 & 0
A80 0
D(A, A80 ) =
0.04$ 0
A80 <$ $ A ! "
4 ( / .
6 ($ > ?1@1E & (
4 ( TA & TCk % ' $ 0
' A & Ck $ 0
= mn/(k(m + n)) = 512 ∗ 512/(80 ∗ 2 ∗ 512) =
512/160 ≈ 3
- $ <
% 3 0
W(A) @ /2 0
0
/& 0
k $ 3
7 % <
$ ( $ >
0
. ;"$ = D(A, A)
& . 6 AF D(A, Ck ) 0
C
k ) 0
& $ & D(A,
. 6 A 7 ? B> 3
TA
1
1
1
a$v &h
3m n/4
( a1 & h1
m n/2
1
9 a
m n/4
; a2 $ v 1 & h1
9m n/16
, a2 $ h2 $ v 1 & h1
5m n/8
D(A, A)
0.013
0.035
0.043
0.070
0.058
k )
D(A, C
0.048
0.052
0.82
0.25
0.069
$ # ' k = 80
C
k )
D(A,
0.047
0.042
0.82
0.25
0.059
# (
A )*"+ & C80 ,
- (
A )*"+ & C80 .
4 $ ( & , 3 (" 0
% 0
%
9 & ; 3 9" 0
3
W(A) 2 $ 0
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. W(Ck ) : 0
D(A, Ck )
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k )$ $ $ 0
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/ DV S 6$ 3 Ck 0
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k$ &
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CK .
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