Download Semi-Supervised with Green`s Function

Learning with Green’s Function with
Application to Semi-Supervised Learning
and Recommender System
----Chris Ding, R. Jin, T. Li and H.D. Simon.
A Learning Framework using Green’s Function and
Kernel Regularization with Application to
Recommender System. KDD’07.
Outline




Green’s Function
Graph-Based Semi-supervised Learning
with Green’s Function
Item-Based Recommendation Using
Green’s Function
Extension
Green’s Function

Green’s Function

Given a weighted graph G=(V,E),
1
0.5
2
0.25
0.8
0.1
4
0.6 5

W=
0.2 0.8 0.5 0 
 1


0.2
1
0.25
0.1
0


 0.8 0.25
1
0 0.4 


0.5
0.1
0
1
0.6


 0
0
0.4 0.6 1 

D=
0
0 0
 2.5 0


0
0 0
 0 1.55
 0
0 2.45 0 0 


0
0
0
2.2
0


 0
0
0
0 2 

0.2
3
0.4
The Graph Laplacian matrix L= D-W.
Green’s Function

Green’s Function

Defined as the inverse of L = D-W with zeromode discarded.
Lvk  k vk , 0  1  2  ...  n
T
n
v
v
1
G*  L 1 
 i i
( D  W ) i 2 i
discard 1  0
Semi-Supervised with Green’s Function

Green’s Function

Interpreted as an electric resistor network
1
wij  I ij  1/ rij
2
w23
3
4
5

voltage : 1
rij  (ei  e j )T G (ei  e j ),
G  ( D  W )  1
ei  (0,..., 0,1, 0,..., 0)
Viewed as a similarity metric on a graph
Semi-Supervised with Green’s Function

Label Propagation

l
l
n
{
x
}
{
y
}
{
x
}
Labeled data
i i 1 &
i i 1 , unlabeled data
i i l 1
labeled data

Label Propagation
For 2-class problems:
l
y j  sign G ji yi , l  j  n
i 1
unlabeled data
For k-class problems:
l

1, k  arg max k  G ji yik
y jk  
,l  j  n
i 1
0, otherwise

Semi-Supervised with Green’s Function

Compared to Harmonic Function
Harmonic Function is an iterative procedure
 Outperforms Harmonic Function
 7 datasets, 10% as labeled data

Recommendation with Green’s Function

Item-based Recommendation
To calculate unknown rating by averaging
rating of similar items by test users
 User-item M  N matrix R,
R pq : u p rates iq
 Item Graph G=(V,E)
typical similarity: cosine similarity, conditional

probability…
Recommendation with Green’s Function
Recommendation with Green’s Function

2

1
0

2
R0  
0

3
3

4
1
3 8 5 0 1 0

0 0 5 0 0 2
2 7 4 7 3 0

4 6 6 8 0 0
1 5 0 5 0 8

2 7 9 0 0 0
6 0 0 0 4 0

5 6 0 0 5 8
2
3
7
4
6
5
R  GR0
T
1
G
( D  W )
T
Recommendation with Green’s Function

Experiments:

Dataset:
Movielens : 943 users; 1682 movies;
ratings from 1 to 5
Training set: 90,570 records
Test set:
9,430 records
Recommendation with Green’s Function

Results compared to traditional methods:
MAE: Mean Absolute Error
 M0E: Mean Zero-one Error

Extension



Combination between semi-supervised
learning and recommendation?
Combine with other recommendation
algorithms?
Improve graph-based semi-supervised
learning with other algorithm?
Discussion and Suggestion
Thank You!