1,2 1 1
!
2
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(
1
3×3 )
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! -
! ./
0 ! 1
1 .
! - 2!
Mn×n 2!
n3 ! 1 n2 (n − 1) 11 3 '454 "
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2!
nlog 7 ! .
Mn×n /
11 n -
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6 9
6
2
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01 2!
.-
! 1
. 11
:
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6!1
.
!6
. ! 2!
1 6 - * !
=< ! 7 #8 - 6 - * ! == 7'8
!
)) Mn×n -
n -
. < > "
? ! nlog 21 ! + log3 21 < log2 7 <
log3 22,
3 1
+@ , 1 . )) - 1 10 0 .! 1 ) . !1 61 758
3
6
)) ! 1! Z . - X 1 Y
Mδ×δ ! σ ! zmn =
δ
p=1
xmp ypn =
σ
r=1
δ
i,j=1
arij xij
δ
brkl ykl
crmn .
∈
+',
k,l=1
. 6
)) 1
1 6 > Ar B r C r ∈ Mδ×δ , 1 ≤ r ≤ σ 1 758 C r 6 !1
61 Ar 1 B r .
Ar 1 B r !2! 1
)) G ≡ (Ar , B r ) 101 . K2δ σ -
K 6
01 6 ' - ; . 101 6 !6
. ! ! ; . 1
.
2
=
1 ; . 6!
1 101 !6
. - 19
.
!6
1 !
. - 6
K = {−1, 0, 1}
6 ' "
1 δ
σ
=
<
<
<
='
==
=<
@ ; "
; A6!
1 ;
7G = 2δ2σ8
7S = 3G8
7N = 2G8
#5
#=<B=5
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<%
==#B'%$
#5
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%'B'%%
4=
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1 61 758 .
G A ≡ A(G) ∈ Mδ ×σ * ! 1! (ρ(Ar )⊗ρ(B r ))T -
ρ(·) - -
! -/
.
&1
* S ∈ Mδ ×δ
- Sαβ = 1 . α = γ(δ2 − 1) + κ β = i(δ − 1) + j κ = k(δ − 1) + j γ = i(δ − 1) + k 1 1 ≤ i, j, k ≤ δ 1 $ -
. G 101 E(A) = A(AT A)−1AT S − S
. AT A !
1 ∞ - 758 C 10 . G 1
f (G) =
-
||E(A)|| = tr(E(A)T E(A)) C ! 0
1 + ||E(A)||
.! +', 61 +=, ! . 1 1 . ! +<, ! 6 !1 ! .! .
. ; 2
1 +, - - 758
4
4
<
2
. . 0*1 arij , brkl 1 ≤ i, j, k, l ≤ δ, 1 ≤ r ≤ σ @ - {−1, 0, 1}
1 1 )) 2! "
? 75 %8 "
.
@ 6 - 6 6 . )) .
. ; 3 × 3
C *11 6 1 75 %8 M3×3 D ! !1 6 1 C 1
1 +
,
. . .
-
! &1
!
* A ∈ Mδ ×σ 1 A1 ∈ Mδ ×q 1
A2 ∈ Mq×σ ! A = A1 A2 6 E(A) =
E(A1 ), 1 !
E(αA) = E(A) α ∈ R ! . 0*1 . p 1 2! ! A 6
!1 6 α = 10p ! 4
4
C 11 @ .
! . )) .
δ = 3 C
1! 1 6 ! 5 1 !
1/ @ 20 . . . 300 11! 11
11! ;1 61 ! 7/#$#$8 :/
1 1 1 ! 1 - 66 $4 1 $$# .- -
!1
' @! 61 gi 9
!1
! 1 gif 1 gim 6 1
6! N (0.5(gif + gim), 0.5|gif − gim|)
= ( 61 "
@! 61 - 19
9
gi 61 6 !.
[gif , gim]
< E! ! :
19
d . gf
1 gm !1 d = sgn f (g f ) − f (g m) g f − gm 9
?
!1 gi1 = gim + γdi 1 gi2 = gif + γdi -
γ !
1 !.
.
[0.1, 0.5]
.- . ! -
1
' @! ! 1 gi !1 1 gi =
gi +μ N (0, 1), -
μ 1 ! 1 6 6!
@! ! .- F f1 f2 1 f3 6 0 !
. 6 11! !
1 ! 2! ! μ !11 1 ! . 6
1 μ = (1 − 0.1α)μ -
α = sgn(2f2 − f1 − f3)
= " ! 1 gi !1 1 gi = gi + κ,
-
κ 1 !.
.
[−10, 10]
< G
! ! - 1 1 -1
@ !
- 6 1- 11! - !
1 ;1 1
1 !1
78 #$$$ 6 11! - 5 . . . 50 1 1
11! .
1 1 #
1 - 1 . ! . 0 1 − 10−10
6 ! - 1 &BB 61 HH" @ F6
1 1 748 6
1-
.
1 . "@3 !
.!
<= 1 %/ 3 I =# @J; . 6 0 11! 1 ! =< 1 ==
! 1 1 :!
' :
=< ! . !
1 :
== ! ! . 0 0.9978 .!1½ :
='
! 6 0 - 6- 0.5
3x3, 22 multiplications
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
Best fitness
Best fitness
3x3, 23 multiplications
1
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
2
3
4
Generation
5
0
0
6
5
x 10
2
4
6
8
Generation
10
12
5
x 10
:!
' @ .
=< 1 == ! ! 0 1! . 6 11!
C 1 2! . !
@ 1 1 ! ?
+, 758 1 H 1 )? +H), 7%8
½
5
C 1 . C 1 1
.!
1 61 G! ; <$$ /6
1 !.
! - 66 $ 1 1! . '$ 1 11/
! -
1 "
758 0 -
1 A 1 11! - 1 .
- 5 1 . '$ 1 .
2! .
'$ - -1 ! .
4.2 × 105 ! .
0 .
=< 1 == ! - :!
= 6 0 !1
! 6 0.11 . !
KB, 3x3, 22 multiplications
0.11
0.1
0.1
0.09
0.09
Best fitness
Best fitness
KB, 3x3, 23 multiplications
0.11
0.08
0.08
0.07
0.07
0.06
0.06
0.05
0
1
2
3
Generation
4
0.05
0
5
5
x 10
1
2
3
Generation
4
5
5
x 10
:!
= .
=< 1 == ! C 1 1 . H) 1 .
3 × 3 .- H) 1! ! !1 *! +F", :
2 × 2 F"
1
'5$$ .
11! F" .6 .
2!
. ; 3 -!1 1 1
9.684 × 107 .
- 1 F" - . '5$$ 1 G! .
-
!
@ H) !1 01 11! . /;
0
C ! 1 .
2! . *
)) 1 1 61 6 1 7=8 C 10 ! . )) Λ XY ∈ Mδ×δ ε(f |X, Y) ≡ f (X, Y) − XY∞ , -
X∞ = max
|Xij | . 3 ; . )) Λ - !1
i
j
1 1
1 1 . ε 1 %' Bi Bj . .
6 . Mδ×δ [R] C .!1 ε = 0.25±0.13 .
- ε = 0.0055 ± 0.0031 .
!
C 1 . @ 01 . ))
.
2!
2!
1 !6
. ! /
. !
! ! - .
- -
6!1 !
6 - !6
6 !1 . 01 6
+ - .-
! , !
- 6 H!
! .! 1 1 ! 2!
23 ! .
2!
. ; 3 - 2! !
6 - !/
11 :
22 ! !
.!1 *
! . 0 0.9978 C 6 01 . !
. * 0 1 .-1 6 1 . 1 3 !
!
! .
!
.!
', ! . !! 1 . 1 =, ! . <, ! .
1 * 1
1 !
1
%
, ! . . 7'8 @ 1 A- "
/ * * ! /
.!1 . !6 2! 7=8 @ 1 .
/
1 * ! /
n2.777 01 1 6! .
- =$$% 7<8 ) K
F-
6!1 .
! * . * !/
%=$<L==5 '444
78 &M/G; )
!
1 !/
+,<''L<< =$$'
7#8 ( C N 1 ) ) F! A ! 6
.
3 × 3 * ! =#4#L5$< '4%5
758 N : 1 G ! !
.
* ! 3 ! '5'L'5# =$$'
78 N D F1
! .
! < * < ! =< ! "
%=+','=5L'=% '45
4
7%8 " H 1 /( ) ! 1! . ! "
? 1 6 ### $ #
'+=,=5L=#' =$'$
748 & "1
1 !
&BB 6
6
.
.
1 ! *
A3& =$'$
7'$8 O "
@! % '<<#L<#5
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