[IEEE 2013 IEEE International Symposium on Information Theory (ISIT) - Istanbul, Turkey...
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Transcript of [IEEE 2013 IEEE International Symposium on Information Theory (ISIT) - Istanbul, Turkey...
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{ }
978-1-4799-0446-4/13/$31.00 ©2013 IEEE
2013 IEEE International Symposium on Information Theory
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ηinηout
K > 1
Pe(ηin, ηout,K) = (γK − γK+1)/(1− γK+1),
γ = ηin/ηout.
N α M{S1 S2 SN} {D1 D2 DM}
{G1 G2 Gα} Si
ηi Zi ∈ {D1, D2, ..., DM}Si Z = [Z1,Z2, · · · ,ZN]
Ri Si
R = [R1,R2, · · · ,RN]Ci
Ci = 1 Ci = 0Gi
C = [C1,C2, · · · ,Cα]
α
C Z
Si
Gj Ri
Li,j(Ri)Pe(ηin, ηout,K)
H(Z)
Ri
i 1 ≤ i ≤ N C
H(Z|R,C)
=H(Z|R,C)
H(Z)=
H(Z)− I(Z;R,C)
H(Z).
C
R
H(Z|R,C) =H(Z)H(Z)
I(Z;R,C)I(Z;R,C)
Q(C1 = c1,C2 = c2, ...,Cα = cα)c1, c2, ..., cα
Q(c1, c2, ..., cα)c1, c2, ..., cα
Q(R1=r1,R2=r2, ...,RN=rN |C,Z)r1, r2, ..., rN
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Z = (z1, z2, ..., zN )C = (c1, c2, ..., cα)
EZ
EZ =α M
=1
N∑i′=1
ηi′
[ α∑i=1
{ ∑R,C:[Ci]=1
ηin(i) ∗ Pe(ηin(i), ηout,K)
+∑
Z,C,R
∑j:Rj∈Gi
ηj ∗ P (Z)Q(C)Q(R|Z,C)Lj,i(Rj)
}
+
M∑l=1
∑Z,C,R
∑j′ :Rj′∈Gi
ηj′ ∗ P (Z)Q(C)Q(R|Z,C)Lj′,Dl(Rj′ )
],
ηin(i) =∑j:Rj∈Gi
∑Z
ηj ∗ P (Z)Q(C)Q(R|Z,C)(1 − Lj,i(Rj)).
Gi
Q(C)Q(R|Z,C)
Q(C)Q(R|Z,C)
minQ(C),Q(R|Z,C):EZ≤D
[H(Z) −H(Z|R,C)],
EZ
Q(R|Z,C) Q(C)DEZ Q(C)
Q(R|Z,C)
Q(R,C|Z) Q(C) ×Q(R|Z,C)Q(R,C|Z)
C Z
Q(C|Z) = Q(C)
minQ∈Λ
∑, ,
P (Z)Q(R,C|Z) logQ(R,C|Z)
q(R,C),
q(R,C) =∑Z
P (Z)Q(R,C|Z),
Λ =
⎧⎪⎪⎨⎪⎪⎩
EZ(Q(R,C|Z)) ≤ D, ∀ Z ,∑,
Q(R,C|Z) = 1, Q(R,C|Z) ≥ 0, ∀ Z ,∑
Q(R,C|Z) =∑,
P (Z)Q(R,C|Z), ∀ Z .
Q∗(R,C|Z)Q∗(C) Q∗(R|Z,C)
Q∗(C) =∑R,Z
Q∗(R,C|Z), Q∗(R|Z,C) =Q∗(R,C|Z)
Q∗(C).
Q∗(C) Q∗(R|Z,C)
Q∗(C) Q∗(R|Z,C) EZ Q∗(C)Q∗(R|Z,C)) ≤ D
Q̂(C)Q̂(R|Z,C)
∑, ,
P (Z)Q̂(C)Q̂(R|Z,C)) logQ̂(C)Q̂(R|Z,C))
q(R,C)
<∑, ,
P (Z)Q∗(C)Q∗(R|Z,C)) logQ∗(C)Q∗(R|Z,C))
q(R,C).
C
Z Q̂(C|Z) = Q̂(C) Z Q̂(R,C|Z) =Q̂(C)Q̂(R|Z,C) EZ(Q(R,C|Z)) ≤ D
∑, ,
P (Z)Q̂(R,C|Z)) logQ̂(R,C|Z))
q(R,C)
<∑, ,
P (Z)Q∗(R,C|Z)) logQ∗(R,C|Z))
q(R,C).
Q∗
Q∗(C), Q∗(R|Z,C)
K > 1
EZ
Q(R,C|Z)f i1
f i1(x) = x ∗ Pe(x, ηout,K), 1 ≤ i ≤ α,
x =∑
j:Rj∈Gi
∑Z
ηj ∗ P (Z)Q(R,C|Z)(1 − Lj,i(Rj)),
Pe(x, ηout,K) dPe
dx≥ 0 d2Pe
dx2 ≥ 0K > 1 K > 1 Pe(x, ηout,K)
x f1(x)x x
Q(R,C|Z) PZ ηj ηout Lj,i f i1(x)
Q(R,C|Z)
Q(R,C|Z) Lj,i(Rj) jηj P (Z)
Q(R,C|Z)
EZ Q(R,C|Z)
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∑, Q(R,C|Z) = 1
∑Q(R,C|Z) =∑
, P (Z)Q(R,C|Z)
I(Z;R,C)Q(R,C|Z) P (Z)
K > 1
Q(R|Z,C) Q(C)
(c1, c2, ..., cα)Q(c1, c2, ..., cα) = 1
Q(R,C|Z) = Q(R|Z,C = c1, c2, ..., cα)
Q(R|Z,C)
Q(R|Z,C) =q(R|C) exp[s
(∑N
i=1 Lq(Ri))]∑
Rq(R|C) exp[s
(∑N
i=1 Lq(Ri))],
q(R|C) =∑Z
P ( )Q(R|Z,C),
Lq(Ri) =∑
j:{Gj∩Ri}∪{Dj∩Ri}
Li,j(Ri) ∗ (ηi/
N∑i′=1
ηi′).
s < 0
s
K
Q(R|Z,C)
f(R) : Q(R|Z,C)−q(R|C) exp[sTZ]∑
R
q(R|C) exp[sTZ)]= 0,
q(R|C) =∑Z
P ( )Q(R|Z,C),
TZ =∂EZ
∂Q(R|Z,C)∗ (1/P ( )).
Q(R|Z,C)Q(R|Z,C)
(n) Q(R|Z,C) nQ(R|Z,C)
(n+1) = (n) −[[∇f ]−1f ]| (n) ,
(n) = [Q(n)( 1| ), Q(n)( 2| ), ..., Q(n)( l| )]T
f = [f( 1), f( 2), ..., f( l)]T
i ⊂ R ⊂ Z
Q(R,C|Z) = Q(C) × Q(R|Z,C)Q(R|Z,C)
Q(C)
Q(C) =
∏R,Z
[q(R,C)
Q(R|Z,C)
]P (Z,R|C)exp[sΦ(C)
]
∑C
( ∏R,Z
[q(R,C)
Q(R|Z,C)
]P (Z,R|C)exp[sΦ(C)]
) ,
q(R,C) =∑C,Z
P ( )Q(R|Z,C)Q(C),
P (Z,R|C) = P ( )Q(R|Z,C),Φ(C) =∂EZ
∂Q(C).
Q(C)
K = 5
P (Z1 = Di,Z2 = Dj) = 1/4 i, j1 2
10−5 10−2
(C1, C3), (C1, C2, C3), (C1, C2, C3, C4)
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C2
S1
{S1, C2, D1} 1 S2
{S2, C4, D2}S1 S2
D1 D2
K
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