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),

W=

D=

The Graph Laplacian matrix L= D-W.

1 2

43

5

0.2

0.25

0.40.6

0.50.8

0.1

1 0.2 0.8 0.5 00.2 1 0.25 0.1 00.8 0.25 1 0 0.40.5 0.1 0 1 0.60 0 0.4 0.6 1

2.5 0 0 0 00 1.55 0 0 00 0 2.45 0 00 0 0 2.2 00 0 0 0 2

Green’s Function Green’s Function

Defined as the inverse of L = D-W with zero-mode discarded.

discard

* 1

2

1( )

Tni i

i i

v vG L

D W

,k k kLv v 1 20 ... n

1 0

Semi-Supervised with Green’s Function

Green’s Function Interpreted as an electric resistor network

1 2

3

5

4

: 1voltage 23w

1/ij ij ijw I r

1

( ) ( ),

( )(0,...,0,1,0,...,0)

Tij i j i j

i

r e e G e e

G D We

Viewed as a similarity metric on a graph

Semi-Supervised with Green’s Function

Label Propagation Labeled data & , unlabeled data

labeled data unlabeled data

For 2-class problems: For k-class problems:

1{ }li ix 1{ }li iy 1{ }ni i lx

1

,l

j ji ii

y sign G y l j n

Label Propagation

1

1, argmax,

0,

l

k ji ikijk

k G yy l j n

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 matrix R, : rates Item Graph G=(V,E) typical similarity: cosine similarity, conditional

probability…

M N

pqR pu qi

Recommendation with Green’s Function

Recommendation with Green’s Function

0

2 3 8 5 0 1 01 0 0 5 0 0 20 2 7 4 7 3 02 4 6 6 8 0 00 1 5 0 5 0 83 2 7 9 0 0 03 6 0 0 0 4 04 5 6 0 0 5 8

R

12 3

456

7 1( )

GD W

0T TR GR

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!