Similarity-based Regularized Latent Feature Model for Link Prediction

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A Î {0, 1} N × M ( n = | V | , m = | W | )

Ç

n

m

n

m

∑ ∑ l(Aij, A ij()) + ∑ ∑ l(Aij, Aij ())S ij + ()

O=

i = 1j = 1

i = 1j = 1

l(., .)

A ij()

l(A ij , A ij ()) = | | A − A | | 2F =

∑ ij(Aij − Aij ) 2

∑ ijAij log

l(A ij, A ij ()) = D(A | | A ) =

A ij A ij

− A ij + A ij

Lr

Precision =

L

¢ ²

AUC =

n¢ + 0.5n² n

s CN xy = | (N(x) Ç N(N(y))) È (N(y) Ç N(N(x)))|

s CN xy

s JC xy =

|N(x) È N(y)|

∑ z Î ( N ( x ) Ç N ( N ( y ) ) ) È ( N ( y ) Ç N ( N ( x ) ) ) log

s AxyA =

s RA xy =

1 2 | N(z)|

1

∑ z Î ( N ( x ) Ç N ( N ( y ) ) ) È ( N ( y ) Ç N ( N ( x ) ) ) |N(z)|

s PA xy = | N(x) | ⋅ | N(y)|

CN LCL s CAR = s ⋅ s xy xy xy

s LCL

s CAR xy

s CJC xy =

A s CA xy

|N(x) È N(y)|

|(z)| = ∑z Î ( N( x ) Ç N( N( y) ) ) È ( N( y) Ç N ( N( x) ) ) ) log 2 | N(z)|

s CRA xy

|(z)| = ∑zÎ ( N( x) Ç N( N( y) ) ) È ( N( y ) Ç N( N ( x) ) ) |N(z)|

(

CA R CAR CAR s CPA = e(x) ⋅ e(y) + e(x) ⋅ s + e(y) ⋅ s + s xy xy xy xy

=

)

2

1 2

n

(

m

)

K

2 1 R = ∑ ∑ A ij − ∑ x ik ⋅ y kj ⋅ S ij 2 i = 1j = 1 k=1

∑K k = 1x ik ⋅ y kj

min O 1(x, y)

=

s.t.

1 A 2

− XY 2F + R + XY

X ≥ 0, Y≥0





1 XY = min X , Y(X 2F + Y 2F ) 2

min O(x, y)

=

1 n m ∑ ∑ 2 i= 1 j=1 1

(

(

A ij − ∑ K k = 1 x ik ⋅ y kj

) ( 1

+ 2 ∑ i ∑ p x 2ip + 2 ∑ j ∑ qy 2qj s.t.

)

2

(

1

K + 2 ∑ ni= 1 ∑ m A − ∑ ij j=1 k=1 x ik ⋅ y kj

)

)

x ik ≥ 0, y kj ≥ 0



L

=

+

∂L ∂x ik

( ) (∑ ∑ x ) + ( ∑ ∑ y ) + ∑ ∑

1 n m ∑ ∑ 2 i= 1 j=1 1 2

i

2 p ip

A ij − ∑ K k = 1 x ik ⋅ y kj 1 2

j

2 q qj

2



(

1

K + 2 ∑ ni= 1 ∑ m A − ∑ ij j=1 k = 1x ik ⋅ y kj

i

kikx ik +

)

2

∑ k ∑ jkjy kj

= − ∑ j[(1 + S ij) ⋅ A ij ⋅ y kj ] + ∑ j[(1 + S ij) ⋅ ( ∑ kx ik ⋅ y kj) ⋅ y kj ] + x ik + ik = −(AY T ) ik − [(S ⋅ A)Y T] ik + [(XY)Y T] ik + [S ⋅ (XY)Y T] ik + X ik + ik

∂L ∂y kj

= − ∑ i[(1 + S ij) ⋅ A ij ⋅ x ik ] + ∑ i[(1 + S ij) ⋅ ( ∑ kx ik ⋅ y kj) ⋅ x ik ] + y kj + kj = −(X TA) kj − [X T (S ⋅ A)] kj + (X TXY) kj + [X T[S ⋅ (XY)]] kj + Y kj + kj kjy kj

=0

− (AY T) ikx ik − [(S ⋅ A)Y T] ikx ik + [(XY)Y T] ikx ik + [S ⋅ (XY)Y T] ikx ik + X ikx ik = 0 − (X TA) kjy kj − [X T(S ⋅ A)] kjy kj + (X TXY) kjy kj + [X T[S ⋅ (XY)]] kjy kj + Y kjy kj = 0

x ik ← x ik ⋅

y kj ← y kj ⋅

[AY T + (S ⋅ A)Y T] ik [XYY T + [S ⋅ (XY)]Y T + X] ik [X TA + X T (S ⋅ A)] kj [X TXY + X T[S ⋅ (XY)] + Y] kj

⋅ S ij

2

⋅ S ij

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Similarity-based Regularized Latent Feature Model for Link Prediction

A Î {0, 1} N × M ( n = | V | , m = | W | ) Ç n m n m ∑ ∑ l(Aij, A ij()) + ∑ ∑ l(Aij, Aij ())S ij + () O= i = 1j = 1 i = 1j = 1 l(., .) A ij...

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