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MissSVM
Multi Instance Learning by Semi-Supervised SVM
cc3736 Chao CHEN
lj2351 Lina JIN
MIL(Multi Instance Learning) problem
1
positive
-1
-1
-1
negative
-1
-1
1
-1
positive
1
-1-1-1
negative
-1
-1
-1
-1
-1
-1 -1
…
SSL (Semi Supervised Learning) problem
negative positive
1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1 ?
?
?
?
?
?
?
?
?
labeled instances unlabeled instances
… …
— Predict label of bags
— Predict label of instances
bag
instance
MissSVM Train
INPUT X1,y1( ), X2,y2( ),..., Xm,ym( ){ } where Xi = xi1,xi2,...,xini
{ } yi = +1 if ∃xik ∈Xi → xik = 1
1:REORDER bag list
X1
−
, X2
−
,..., Xq
−
, Xq+1
+
,..., Xq+p−1
+
, Xm
+
{ }
2:MAP into instance list
Lx = x1,1,...,x1,n1
,x2,1,...,xm,1,...,xm,nm
{ }
Given a set of labeled negative instances x1,−1( ), x2,−1( ),..., xTL
,−1( ){ }
of unlabeled instances
and a set
xTL +1,...,xT{ } , to learn a function Fs
: x → {−1, +1}
Subject to: For i = q +1, ... ,m at least one instance in
{xsi
, ... ,xei
} is +1
total number of instances in training setT
TL
total number of instances labeled with -1
q total number of negative bags
m total number of bags
si
index in Lx of the first instance belonging to ith bag
index in Lx of the last instance belonging to ith bagei
3:Learn a Semi-Supervised SVM
-1 otherwiseλ, γ , δ
X = x1,x2,...,xn{ }
+1 if ∃xi ∈X, s.t. Fs
xi( )= +1
INPUT
RETURN
OUTPUT y ∈ −1, 1{ }
−1 otherwise
MissSVM Predict
Fs
Learn a Semi Supervised SVM
Optimization problem for popular semi-supervised SVM
min
f
1
2
|| f ||H
2
+λ H1 yt f xt( )( )
t=1
TL
∑ +δ D f xt( )( )
t=TL +1
T
∑
f : x → R
|| f ||H norm of Reproducing Kernel Hilbert Space of f
H1 z( )= max 0, 1− z{ } Hinge Loss
D z( )= min H1 z( ), H1 −z( ){ } A non-convex hat shape loss functionD z( )= min H1 z( ), H1 −z( ){ }
1.Positive Constrains
2.Dimension of space
For i = q +1, ... ,m , at least one instance in {xsi
, ... ,xei
} is +1.
To reduce the optimization problem from a possibly infinite-dimensional
space to a finite-dimensional space
Considerations:
Modified to be a CCCP(Constrained Concave-Convex Procedure)
min
α,η,θ,ε,b
1
2
α 'Kα + λη'1+γθ '1+δ min ε,ξ( )'1
s.t.
(−1)(k't α + b)+ηt ≥1, ηt ≥ 0, t = 1,2,...,TL;
max
t=si ,...,ei
(k't α + b)+θi−q ≥1,θi−q ≥ 0, i = q +1,...,m;
(k't α + b)+ εt−TL
≥1,εt−TL
≥ 0, t = TL +1,...,T;
(−1)(k't α + b)+ξt−TL
≥1,ξt−TL
≥ 0, t = TL +1,...,T.
⎧
⎨
⎪
⎪⎪
⎩
⎪
⎪
⎪
η = η1,...,ηTL
⎡⎣ ⎤⎦' slack variables for the error on instances of negative bags
θ = θ1,...,θp
⎡⎣ ⎤⎦' slack variables for the error on positive bags
ε = ε1,...,εTU
⎡⎣ ⎤⎦'ξ = ξ1,...,ξTU
⎡⎣ ⎤⎦' slack variables for the error on instances of positive bags
λ,γ ,δ user defined parameters to trade off complexity with errors
K a TxT kernel matrix
Fs
Learn a Semi Supervised SVM
Revised to be a standard QP
min
α,η,θ,ε,b
1
2
α 'Kα + λη'1+γθ '1+δ ∂ min εa
,ξa
( )'1( )(
ε
ξ
)
s.t.
(−1)(k't α + b)+ηt ≥1, ηt ≥ 0, t = 1,2,...,TL;
βit
a
t=si
ei
∑ k't α + b +θi−q ≥1,θi−q ≥ 0, i = q +1,...,m;
(k't α + b)+ εt−TL
≥1,εt−TL
≥ 0, t = TL +1,...,T;
(−1)(k't α + b)+ξt−TL
≥1,ξt−TL
≥ 0, t = TL +1,...,T.
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
Fs
Learn a Semi Supervised SVM
βit =
= 0, if kt 'α ≠ max
r=si ,...,ei
kr 'α
=
1
na
,otherwise
⎧
⎨
⎪⎪
⎩
⎪
⎪
∂ max
t=si ,...,ei
kt 'α( )= βit kt '
t=s i
ei
∑
Dataset: Musk
Data generated in the research of drug activity prediction
Algorithm Musk1 Musk2
MissSVM 87.6 82.3
mi-SVM 87.4 83.6
MI-SVM 77.9 84.3
Diverse Density 88.9 82.5
Table 1. Predictive accuracy(%) on the Musk data

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AML

  • 1. MissSVM Multi Instance Learning by Semi-Supervised SVM cc3736 Chao CHEN lj2351 Lina JIN
  • 2. MIL(Multi Instance Learning) problem 1 positive -1 -1 -1 negative -1 -1 1 -1 positive 1 -1-1-1 negative -1 -1 -1 -1 -1 -1 -1 … SSL (Semi Supervised Learning) problem negative positive 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ? ? ? ? ? ? ? ? ? labeled instances unlabeled instances … … — Predict label of bags — Predict label of instances bag instance
  • 3. MissSVM Train INPUT X1,y1( ), X2,y2( ),..., Xm,ym( ){ } where Xi = xi1,xi2,...,xini { } yi = +1 if ∃xik ∈Xi → xik = 1 1:REORDER bag list X1 − , X2 − ,..., Xq − , Xq+1 + ,..., Xq+p−1 + , Xm + { } 2:MAP into instance list Lx = x1,1,...,x1,n1 ,x2,1,...,xm,1,...,xm,nm { } Given a set of labeled negative instances x1,−1( ), x2,−1( ),..., xTL ,−1( ){ } of unlabeled instances and a set xTL +1,...,xT{ } , to learn a function Fs : x → {−1, +1} Subject to: For i = q +1, ... ,m at least one instance in {xsi , ... ,xei } is +1 total number of instances in training setT TL total number of instances labeled with -1 q total number of negative bags m total number of bags si index in Lx of the first instance belonging to ith bag index in Lx of the last instance belonging to ith bagei 3:Learn a Semi-Supervised SVM -1 otherwiseλ, γ , δ
  • 4. X = x1,x2,...,xn{ } +1 if ∃xi ∈X, s.t. Fs xi( )= +1 INPUT RETURN OUTPUT y ∈ −1, 1{ } −1 otherwise MissSVM Predict
  • 5. Fs Learn a Semi Supervised SVM Optimization problem for popular semi-supervised SVM min f 1 2 || f ||H 2 +λ H1 yt f xt( )( ) t=1 TL ∑ +δ D f xt( )( ) t=TL +1 T ∑ f : x → R || f ||H norm of Reproducing Kernel Hilbert Space of f H1 z( )= max 0, 1− z{ } Hinge Loss D z( )= min H1 z( ), H1 −z( ){ } A non-convex hat shape loss functionD z( )= min H1 z( ), H1 −z( ){ } 1.Positive Constrains 2.Dimension of space For i = q +1, ... ,m , at least one instance in {xsi , ... ,xei } is +1. To reduce the optimization problem from a possibly infinite-dimensional space to a finite-dimensional space Considerations:
  • 6. Modified to be a CCCP(Constrained Concave-Convex Procedure) min α,η,θ,ε,b 1 2 α 'Kα + λη'1+γθ '1+δ min ε,ξ( )'1 s.t. (−1)(k't α + b)+ηt ≥1, ηt ≥ 0, t = 1,2,...,TL; max t=si ,...,ei (k't α + b)+θi−q ≥1,θi−q ≥ 0, i = q +1,...,m; (k't α + b)+ εt−TL ≥1,εt−TL ≥ 0, t = TL +1,...,T; (−1)(k't α + b)+ξt−TL ≥1,ξt−TL ≥ 0, t = TL +1,...,T. ⎧ ⎨ ⎪ ⎪⎪ ⎩ ⎪ ⎪ ⎪ η = η1,...,ηTL ⎡⎣ ⎤⎦' slack variables for the error on instances of negative bags θ = θ1,...,θp ⎡⎣ ⎤⎦' slack variables for the error on positive bags ε = ε1,...,εTU ⎡⎣ ⎤⎦'ξ = ξ1,...,ξTU ⎡⎣ ⎤⎦' slack variables for the error on instances of positive bags λ,γ ,δ user defined parameters to trade off complexity with errors K a TxT kernel matrix Fs Learn a Semi Supervised SVM
  • 7. Revised to be a standard QP min α,η,θ,ε,b 1 2 α 'Kα + λη'1+γθ '1+δ ∂ min εa ,ξa ( )'1( )( ε ξ ) s.t. (−1)(k't α + b)+ηt ≥1, ηt ≥ 0, t = 1,2,...,TL; βit a t=si ei ∑ k't α + b +θi−q ≥1,θi−q ≥ 0, i = q +1,...,m; (k't α + b)+ εt−TL ≥1,εt−TL ≥ 0, t = TL +1,...,T; (−1)(k't α + b)+ξt−TL ≥1,ξt−TL ≥ 0, t = TL +1,...,T. ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ Fs Learn a Semi Supervised SVM βit = = 0, if kt 'α ≠ max r=si ,...,ei kr 'α = 1 na ,otherwise ⎧ ⎨ ⎪⎪ ⎩ ⎪ ⎪ ∂ max t=si ,...,ei kt 'α( )= βit kt ' t=s i ei ∑
  • 8. Dataset: Musk Data generated in the research of drug activity prediction Algorithm Musk1 Musk2 MissSVM 87.6 82.3 mi-SVM 87.4 83.6 MI-SVM 77.9 84.3 Diverse Density 88.9 82.5 Table 1. Predictive accuracy(%) on the Musk data