Journal name. Page 2 of 21. For Peer Review. A r ( )q r ( ) =f r ( ) A h f h.

For Peer Review

A^r(✓)q^r(✓) =f^r(✓)

A^h f^h 2

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For Peer Review

˜ q^r(✓) : R^Np7!R^Nr q^r2R^Nr

M q= (q1, . . . , q)^T

q^r= (q^r₁, . . . , q^r_M^r)^T

M^r M^r⌧

M

q M^r M

q=W q^r W 2R^M⇥Mr q=G(q^r) G

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For Peer Review

F(✓) = 0

Ah(✓)qh(✓) =fh(✓),

Ah(✓) =PN A

n=1cAn(✓)Ah n fh(✓) =PN f

n=1cf n(✓)fh

n

qh(✓m)

V

Ar

n=VTAh nV fr

n=VTfh n

✓

Ar(✓) =PN A

n=1cAn(✓)Ar n fr(✓) =PN f

n=1cf n(✓)fr

n

Ar(✓)qr(✓) =fr(✓)

qh(✓) =V qr(✓)

A^h_n,f^h_n,A^r_n,f^r_n ✓

F(✓) = 0

Ah(✓)qh(✓) =fh(✓)

uh(✓m)

V

qr=VTqh

˜ qr(✓) :

˜

qr(✓)⇡qr(✓)

✓

˜ qr(✓)

qh(✓) =Vq˜r(✓)

P=n

✓¹, . . . ,✓^Mso S2R^Mh⇥Ms

S=h

˜

q^h ✓¹ . . . q˜^h⇣

✓^Ms⌘ i , M^h

S=U⌃V^T,

U = [⇣₁|. . .|⇣_M^h]2R^Mh⇥Mh, V = [ ₁|. . .| M^s]2R^Ms⇥Ms,

⌃= ( 1, . . . , M^r)2R^M⇤⇥M⇤. 2

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For Peer Review

⇣₁, . . . ,⇣_M^h S

1, . . . , _M^s S 1, . . . , M^⇤

S M^⇤= min M^h, M^s

1 2. . . M^⇤ 0

M^h

O(10⁷)

S=U^⇤⇤⌃^⇤⇤V^T

U^⇤⇤= [⇣₁|. . .|⇣_M⇤⇤]2R^Mh⇥M⇤⇤,

⌃^⇤⇤= ( 1, . . . , ^M⇤⇤)2R^{M⇤⇤⇥M⇤⇤}. M^⇤⇤= (S)min(M^h, M^s)

S=U^⇤⇤C C =

⌃^⇤⇤V^T 2R^r⇥Ms U^⇤⇤

C C =U^⇤⇤TS

U^⇤⇤

✓

M^rM^⇤⇤ Ub

Ub = [⇣₁|. . .|⇣_Mr]2R^Mh⇥Mr V= nW 2R^Mh⇥Mr :W^TW =Io

k

M^s

X

m=1

˜

q^h(✓^m) UbUb^T˜q^h(✓^m)

L²!²

= min

W2V M^s

X

m=1

✓

˜

q^h(✓^m) W W^T˜q^h(✓^m) ^L

2◆²

=

MX^⇤⇤

m=M^r+1 2m,

PM^r m=1 2

PM^⇤⇤ m m=1 2

m

1 ²,

kS UbUb^TSk^F kSk^F  ,

k·k^{F k}

k

{q₀, . . . ,q_m}

X= [q₀· · ·q_{m 1}], Y = [q₁· · ·q_m]

{(x1,y₁), . . . ,(xm,y_m)}

X= [x1· · ·xm], Y = [y₁· · ·y_m]

xk =q_{k 1},y_k=q_k

A=Y X⁺,

X⁺ X A

AX =Y

AX =Y kAk^F

kAX Yk^F k·k A n

A

{(x1,y₁), . . . ,(xm,y_m)}

X Y

X X=U⌃V^T A˜ =U^TY V⌃ ¹

A˜ A!˜ = !

= ¹Y V⌃ ¹! 2

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For Peer Review

( , )

A q_k

q_k = Xm j=1

kj j.

m _j

j

h:q7!q˜ q˜⇡q q7!q^r=Gen(q;!en)

ˆ q q^r7!q˜=Gde(q^r;!de)

(!en,!de)

(!^⇤_en,!^⇤_de) = arg min

!en,!deL(˜q,q), L

G^en G^de

Gen=W^Tq,Gde=W q^r

W^⇤= arg min

W q W W^Tq ,

W

O(10³)

O(10⁷)

10⁷ 10 10⁸

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For Peer Review

F(✓) = 0 q^r

⌫_mⁱ ⌫_m^hl ⌫_m^o

m l

X2R^Ni!Y 2R^No Nⁱ N^o

⌫ⁱ₁

⌫ⁱ₂

⌫ⁱ₃

⌫ⁱ₄

⌫₁^h1

⌫₂^h1

⌫₃^h1

⌫₄^h1

⌫₅^h1

⌫₆^h1

⌫₁^h2

⌫₂^h2

⌫₃^h2

⌫₄^h2

⌫₅^h2

⌫₁^o

⌫₂^o

⌫₃^o

8>

>>

>>

>>

<

>>

>>

>>

>:

⌫ⁱ = X,

⌫^h1 = ^h1⇣

W^h1⌫ⁱ+B^h1⌘ ,

⌫^hl = ^hl⇣

W^hl⌫^{hl 1}+B^hl⌘

, l= 2, . . . , N^L,

⌫^o = ^o⇣

W^o⌫^NL+B^o⌘ , Yˆ = ⌫^o,

W B

(z) = max(0, z)

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For Peer Review

N^L N^hl

Nⁱ N^o

N^L

N^hl

F:R^ki!R

kⁱ cⁱ

k^o 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=⇥

cⁱ₁ . . . cⁱ_M⇤ d

K dm=D cⁱ_m Kmn=K cⁱ_m,cⁱ_n

P c^o|Cⁱ =N(c^o|d,K),

P N c^o

M c^o_m=F cⁱ_m , m= 1, . . . , M

Cⁱ_⇤=⇥

cⁱ_⇤,1 . . . cⁱ_⇤,M⇤⇤

m= 1, . . . , M_⇤

✓ c^o c^o_⇤

◆

⇠N

✓✓ d d_⇤

◆ ,

✓ K K_⇤ K^T_⇤ K_⇤⇤

◆◆

,

P c^o_⇤|Cⁱ_⇤ =N(c^o_⇤|d_⇤,K_⇤),

d_⇤,m=D cⁱ_⇤,m K_⇤,mn=K cⁱ_⇤,m,cⁱ_n K_⇤⇤,mn= K cⁱ_⇤,m,cⁱ_⇤,n

P c^o_⇤|Cⁱ_⇤,Cⁱ,c^o =N(c^o_⇤|d^c_⇤,K^c_⇤),

d^c_⇤=d_⇤+K^T_⇤K ¹(c^o d), K^c_⇤=K_⇤⇤ K^T_⇤K ¹K_⇤.

c^✏_m=F cⁱ_m +✏, ✏⇠N 0, ² , m= 1, . . . , M.

✓ c^✏ c^o_⇤

◆

⇠N

✓✓ d d_⇤

◆ ,

✓ K^✏ K_⇤ K^T_⇤ K_⇤⇤

◆◆

, K^✏=K+ ²I

P c^o_⇤|Cⁱ_⇤,Cⁱ,c^✏ =N(c^o_⇤|d^✏_⇤,K^✏_⇤),

d^✏_⇤=d_⇤+K^T_⇤ (K^✏) ¹(c^✏ d) K^✏_⇤=K_⇤⇤ K^T_⇤ (K^✏) ¹K_⇤.

D cⁱ = 0 d^✏_⇤

d^✏_⇤=K^T_⇤ (K^✏) ¹c^✏. d^✏_⇤2R^M⇤

Cⁱ_⇤

! 2

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For Peer Review

M

c^o_j = XM d=1

aj,dud, j= 1,2,· · ·, M , cj

M ud⇠

GP(D^d(·),K^d(·,·)) ad

ud

K(x,x⁰) = XM d=1

ada^T_dK^d(x,x⁰).

c^o_L c^o_H

c^o_H =⇢u1+u2, c^o_L=u1.

(Cⁱ_L,c^✏_L) (Cⁱ_H,c^✏_H)

✏L⇠N(0, ²_L)

✏H⇠N(0, ²_H)

c^o_H⇤ |Cⁱ_H,c^✏_H,Cⁱ_L,c^✏_L⇠GP(D⇤(·),K⇤(·,·)),

D⇤(x) = h(x) +g(x,Cⁱ_H)[g(Cⁱ_H,Cⁱ_H) + ²_HI] ¹

·(c^✏_H h(Cⁱ_H)),

K⇤(x,x⁰) = g(x,x⁰) g(x,Cⁱ_H)[g(Cⁱ_H,Cⁱ_H) + ²_HI] ¹

·g(Cⁱ_H,x⁰), h(x) = ⇢D¹(x) +D²(x),

g(x,x⁰) = K²(x,x⁰) +⇢²{K¹(x,x⁰) K¹(x,Cⁱ_L)

·[K¹(Cⁱ_L,Cⁱ_L) + ²_LI] ¹K¹(Cⁱ_L,x⁰)}.

⇢

ˆ

q ✓

SH

ˆ

q(✓) =SHˆc(✓),

cˆ

ˆ

c(✓) = arg min

v kq_L(✓) SLvk²,

q_L SL

ˆ

c(✓) = (S^T_LSL) ¹S^T_Lq_L(✓). 2

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For Peer Review

SL

SL

P SLP =QR. P

q(x, t;✓)⇡a(x, t;✓) + XK k=1

XM 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)

✓ 2

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For Peer Review

4⇥4 3⇥5 3⇥5

[ 1,1]²

Eq˙ =Aq+H(q⌦q), E A2R^Mh⇥Mh H2R^Mh⇥(Mh)2

W 2R^Mh⇥Mr S 2R^Mh⇥M⇤

Eˆq˙ˆ= ˆAˆq+ ˆH(ˆq⌦q)ˆ ,

Eˆ =W^TEW Aˆ =W^TAW Hˆ = W^TH(W ⌦W)

S S˙

Sˆ =W^TS, S˙ˆ =W^TS˙ , Eˆ Aˆ Hˆ

ˆ min

E2R^{M r⇥M r},A2Rˆ ^{M r⇥M r},H2Rˆ ^{M r⇥M r2} M^⇤

X

k=1

kEˆs˙ˆk Aˆˆsk H(ˆˆ sk⌦ˆsk)k²,

sk ˆsk k 2

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For Peer Review

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

For Peer Review

(Ub^o)1 (Ub^o)2 (Ub^o)3

(Ub^o)7 (Ub^o)19 (Ub^o)32

(Ub^o)m m Ub^o

0.2 0.4 0.6 0.8 0.0 2.0 0.0

30.0 60.0

✓1 ✓2

co 1 14.0

37.0 60.0 c^o1

0.2 0.4 0.6 0.8 0.0 2.0 5.00

0.00 5.00

✓1 ✓2

co 2

6.00 1.50 3.00 c^o2

0.2 0.4 0.6 0.8 0.0 2.0 5.00

0.00 5.00

✓1 ✓2

co 3

6.20 1.00 4.20 c^o3

0.2 0.4 0.6 0.8 0.0 2.0 2.50

0.50 1.50

✓1 ✓2

co 7

1.70 0.10 1.50 c^o₇

0.2 0.4 0.6 0.8 0.0 2.0 0.90

0.25 0.40

✓1 ✓2

co 19

0.42 0.06 0.54 c^o₁₉

0.2 0.4 0.6 0.8 0.0 2.0 0.40

0.00 0.40

✓1 ✓2

co 32

0.48 0.15 0.18 c^o₃₂

✓1

✓2

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For Peer Review

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

For Peer Review

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

For Peer Review

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

For Peer Review

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

For Peer Review

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

For Peer Review

http://mc.manuscriptcentral.com/(site) 2

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