(1)For Peer Review
Ar(✓)qr
(✓) = fr
(✓)
Ah fh
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For Peer Review
˜
qr(✓) :
RNp
7! RNr
qr
2 RNr
M q = (q1, . . . , q)T
qr
= (qr
1, . . . , qrMr)T
Mr Mr⌧
M
q
Mr
M
q = W qr
W
2 RM⇥Mr
q =G (qr
)
G
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For Peer Review
F(✓) = 0
Ah (✓)qh (✓) = f h (✓),
Ah (✓) =PNA
n=1 cA
n (✓)Ah
n
f h (✓) =PNf
n=1 cf
n (✓)fh
n
qh (✓m)
V
Ar
n = VT Ah
n V f r
n = VT fh
n
✓
Ar (✓) =PNA
n=1 cA
n (✓)Ar
n
f r (✓) =PNf
n=1 cfn (✓)frn
Ar (✓)qr (✓) = f r (✓)
qh (✓) = V qr (✓)
Ah
n, fhn, Arn, frn ✓
F(✓) = 0
Ah (✓)qh (✓) = f h (✓)
uh (✓m)
V
qr = V T qh
˜
qr (✓) :
˜
qr (✓)⇡ qr (✓)
✓
˜
qr (✓)
qh (✓) = V ˜qr (✓)
P =n✓1, . . . , ✓Mso
S2 RMh⇥Ms
S =h˜qh ✓1 . . . ˜qh⇣✓Ms⌘ i,
Mh
S = U ⌃VT,
U = [ ⇣1| . . . |⇣Mh]2 RM
h
⇥Mh
,
V = [ 1| . . . | Ms]2 RM
s
⇥Ms
,
⌃ = ( 1, . . . , Mr)2 RM
⇤
⇥M⇤
.
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For Peer Review
⇣1, . . . , ⇣Mh S
1, . . . , Ms S 1, . . . , M⇤
S M⇤= min Mh, Ms
1 2. . . M⇤ 0
Mh
O(107)
S = U⇤⇤⌃⇤⇤VT
U⇤⇤= [ ⇣1| . . . |⇣M⇤⇤]2 RM
h
⇥M⇤⇤
,
⌃⇤⇤= ( 1, . . . , M
⇤
⇤
)
2 RM⇤⇤⇥M⇤⇤.
M⇤⇤= (S) min(Mh
, Ms
)
S = U⇤⇤C C =
⌃⇤⇤VT 2 Rr⇥Ms
U⇤⇤
C
C = U⇤⇤TS
U⇤⇤
✓
Mr
M⇤⇤
Ub
b
U = [ ⇣1| . . . |⇣Mr]2 RM
h
⇥Mr
V =
n
W 2 RMh⇥Mr
: WTW = Io
k
Ms
X
m=1
˜
qh(✓m) U bbUT˜qh(✓m)
L2!2
= min
W2V
Ms
X
m=1
✓
˜
qh(✓m) W WT˜qh(✓m) L
2◆2
=
M
X⇤⇤
m=Mr
+1
2
m,
PMr
m=1 m2
PM⇤⇤
m=1 m2
1 2,
kS U bbUTS
kF
kSkF ,
k · kF k
k
{q0, . . . , qm}
X = [q0· · · qm 1], Y = [q1· · · qm]
{(x1, y1), . . . , (xm, ym)}
X = [x1· · · xm], Y = [y1· · · ym]
xk = qk 1, yk= qk
A = Y X+,
X+ X A
AX = Y
AX = Y kAkF
kAX YkF k · k
A
n
A
{(x1, y1), . . . , (xm, ym)}
X Y
X X = U ⌃VT
˜
A = UTY V ⌃ 1
˜
A
˜
A! = !
= 1Y V ⌃ 1!
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For Peer Review
( , )
A
qk
qk =
m
X
j=1
k
j j.
m j
j
h : q
7! ˜q q˜
⇡ q
q
7! qr
=G
en(q; !en)
ˆ
q
qr
7! ˜q = Gde(qr; !de)
(!en, !de)
(!⇤en, !⇤de) = arg min
!en,!deL (˜q, q) ,
L
Gen Gde
Gen= WTq,Gde= W qr
W⇤= arg min
W q W W
T
q ,
W
O(103
)
O(107
)
107
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108
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For Peer Review
F(✓) = 0
qr
⌫i
m ⌫mhl ⌫mo
m l
X
2 RNi
! Y 2 RNo
Ni
No
⌫i
1
⌫i
2
⌫i
3
⌫i
4
⌫h1
1
⌫h1
2
⌫h1
3
⌫h1
4
⌫h1
5
⌫h1
6
⌫h2
1
⌫h2
2
⌫h2
3
⌫h2
4
⌫h2
5
⌫o
1
⌫o
2
⌫o
3
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
⌫i
= X,
⌫h1
= h1
⇣
Wh1
⌫i
+ Bh1⌘
,
⌫hl
= hl
⇣
Whl
⌫hl 1
+ Bhl⌘
, l = 2, . . . , NL
,
⌫o
= o⇣
Wo
⌫NL
+ Bo⌘,
ˆ
Y = ⌫o
,
W B
(z) =
max(0, z)
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For Peer Review
NL
Nhl
Ni
No
NL
Nhl
F : Rki
! R
ki
ci
ko
F
D ci
K ci
D ci
=
E F ci
,
K⇣ci
, ´ci⌘
=
E⇣ F ci
D ci ⇣
F⇣´ci⌘
D⇣´ci⌘⌘⌘,
E ⇣ci
, ´ci⌘
2
I ⇥ I
F ci ⇠ G⇣D ci ,
K⇣ci, ´ci⌘⌘.
Ci=⇥ci
1 . . . ciM
⇤
d
K dm=D cim
Kmn=K cim, cin
P co
|Ci =N (co
|d, K) ,
P
N co
M
co
m=F cim , m = 1, . . . , M
Ci
⇤=⇥ci
⇤,1 . . . ci⇤,M⇤
⇤
m = 1, . . . , M
⇤
✓
co
co
⇤
◆
⇠ N
✓✓
d
d⇤
◆
,
✓
K K⇤
KT
⇤ K⇤⇤
◆◆
,
P co
⇤|Ci⇤ =N (co⇤|d⇤, K⇤) ,
d
⇤,m=
D ci
⇤,m K⇤,mn=K ci⇤,m, cin K⇤⇤,mn=
K ci
⇤,m, ci⇤,n
P co
⇤|Ci⇤, Ci, co =N (co⇤|dc⇤, Kc⇤) ,
dc
⇤= d⇤+ KT⇤K 1(co d) ,
Kc
⇤= K⇤⇤ KT⇤K 1K⇤.
c✏m=F cim + ✏, ✏⇠ N 0, 2 , m = 1, . . . , M.
✓
c✏
co
⇤
◆
⇠ N
✓✓
d
d⇤
◆
,
✓
K✏
K
⇤
KT
⇤ K⇤⇤
◆◆
,
K✏= K + 2
I
P co
⇤|Ci⇤, Ci, c✏ =N (co⇤|d
✏
⇤, K✏⇤) ,
d✏
⇤= d
⇤+ KT
⇤ (K✏) 1(c✏
d)
K✏
⇤= K
⇤⇤ KT
⇤ (K✏) 1K
⇤.
D ci
= 0
d✏
⇤
d✏
⇤= KT
⇤ (K✏) 1c✏.
d✏
⇤2 RM
⇤
Ci
⇤
!
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For Peer Review
M
coj =
M
X
d=1
aj,dud, j = 1, 2,· · · , M ,
cj
M ud⇠
GP(Dd(·), Kd(·, ·)) ad
ud
K(x, x0
) =
M
X
d=1
adaTdKd(x, x0) .
co
L coH
coH = ⇢u1+ u2,
coL= u1.
(CiL, c✏L) (C
i
H, c✏H)
✏L⇠ N (0, 2L)
✏H⇠ N (0, 2H)
coH⇤ | CiH, c✏H, CiL, c✏L⇠ GP(D⇤(·), K⇤(·, ·)) ,
D⇤(x) = h(x) + g(x, CiH)[g(C
i
H, C
i
H) + 2HI] 1
· (c✏
H h(CiH)) ,
K⇤(x, x0) = g(x, x0) g(x, CiH)[g(CiH, CiH) + 2HI] 1
· g(CiH, x0) ,
h(x) = ⇢
D1(x) +D2(x) ,
g(x, x0) =
K2(x, x0) + ⇢2{K1(x, x0) K1(x, CiL)
· [K1(CiL, CiL) + 2LI] 1K1(CiL, x0)} .
⇢
ˆ
q ✓
SH
ˆ
q(✓) = SHˆc(✓) ,
ˆ
c
ˆ
c(✓) = arg min
v kqL(✓) SLvk
2
,
qL SL
ˆ
c(✓) = (ST
LSL) 1STLqL(✓) .
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For Peer Review
SL
SL
P
SLP = QR .
P
q (x, t; ✓)
⇡ a (x, t; ✓) +
K
X
k=1
M
X
m=1
↵km(✓) k(x)⇠m(t),
a (x, t; ✓)
↵km(✓)
k
(x) ⇠m
(t)
k,n
k=1,2,...,Kn (⇠m,n)m=1,2,...,Mn
✓n
k
(x)
⇠m
(t)
✓
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For Peer Review
4⇥ 4 3⇥ 5 3⇥ 5
[ 1, 1]2
E ˙q = Aq + H(q⌦ q) ,
E A
2 RMh
⇥Mh
H
2 RMh
⇥(Mh
)2
W 2 RMh
⇥Mr
S 2 RMh
⇥M⇤
ˆ
E ˙ˆq = ˆAˆq + ˆH(ˆq⌦ ˆq) ,
ˆ
E = WTEW A = Wˆ TAW H =ˆ
WTH(W
⌦ W )
S S˙
ˆ
S = WTS , ˙ˆS = WTS ,˙
ˆ
E ˆA Hˆ
min
ˆ
E2RM r⇥Mr
, ˆA2RM r⇥Mr
, ˆH2RM r⇥Mr 2
M⇤
X
k=1
k ˆE ˙ˆsk Aˆˆsk H(ˆˆ sk⌦ ˆsk)k2,
sk ˆsk k
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For Peer Review
( bUo)1 ( bUo)2 ( bUo)3
( bUo)7 ( bU
o
)19 ( bU
o
)32
( bUo)m m Ub
o
0.2 0.4 0.6 0.80.0
2.0
0.0
30.0
60.0
✓1 ✓2
c
o 1 14.0
37.0
60.0
co
1
0.2 0.4 0.6 0.80.0
2.0
5.00
0.00
5.00
✓1 ✓2
c
o 2
6.00
1.50
3.00
co
2
0.2 0.4 0.6 0.80.0
2.0
5.00
0.00
5.00
✓1 ✓2
c
o 3
6.20
1.00
4.20
co
3
0.2 0.4 0.6 0.80.0
2.0
2.50
0.50
1.50
✓1 ✓2
c
o 7
1.70
0.10
1.50
co
7
0.2 0.4 0.6 0.80.0
2.0
0.90
0.25
0.40
✓1 ✓2
c
o 19
0.42
0.06
0.54
co
19
0.2 0.4 0.6 0.80.0
2.0
0.40
0.00
0.40
✓1 ✓2
c
o 32
0.48
0.15
0.18
co
32
✓1
✓2
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For Peer Review
http://mc.manuscriptcentral.com/(site)
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