ICONIP 2010, Sydney, Australia 1

An Enhanced Semi-supervised Recommendation Model Based

on Green’s Function

Dingyan Wang and Irwin KingDept. of Computer Science & Engineering

The Chinese University of Hong Kong

OutlineBackgroundMotivationAn Enhanced ModelExperimental AnalysisConclusion

2ICONIP 2010, Sydney, Australia

Background• Recommendation in Collaborative Filtering

Recommendation

ICONIP 2010, Sydney, Australia 3

Background

• Significance– Consumer Satisfaction– Profit

• Mathematical Form– User-item matrix

complete task– Rating prediction

0

2 3 4 5 ? 1 ?1 ? ? 3 ? 4 2? 2 3 4 4 3 ?2 4 5 1 3 ? ?? 1 5 ? 5 ? 23 2 4 3 ? ? ?.. .. .. .. .. .. ..

R

User

Item

Rating for Prediction

ICONIP 2010, Sydney, Australia 4

Background

• Traditional Recommendation Methods– Memory-based method

• Item-based method, WWW ’01 & SIGIR ’06

• User-based method, SIGIR ’06

– Model-based method• Probabilistic matrix factorization, SIGIR ’07 & 04

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Background

• A Novel View of Recommendation [Green’s function recommendation, KDD ’07 & WWW10]

– Label propagation on a graph

– Label prediction with semi-supervised learning

2

3

54

1

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Motivation

• Higher accuracy in label propagation recommendation

• Importance of graph construction• Accuracy Reduction

– Data Sparsity• Some items have no similarity information

– Information Loss• Similarity in a local view

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An Enhanced Model

• An Enhanced Model Based on Green’s Function

Enhanced Item-Graph Construction

User-Item Rating Matrix

Green’s Function Calculation

0R

Label Propagation

Predicted User-item Matrix '

0R

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An Enhanced Model

• Enhanced Item-Graph Construction– Global similarity between items

• Latent-feature vector similarity– Local similarity between items

• Similarity derived from ratings– Global and local consistent similarity

• Linear combination of global and local similarity

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An Enhanced Model

• Global Similarity Calculation– Latent features extraction

• Probabilistic matrix factorization (PMF), NIPS ’08

R UV

: M*N rating matrix ; : K*N item-latent matrix : M*K user-latent: rating of user i for item j; : indicator to show whether user i rated item j.

R V

2 2

1 1

( | , , ) [ ( | , )]ijIm n

Tij i j

i j

p R U V N R U V

ijR ijI

2 2 200

1 1

1min || || min ( , , ) min ( ) || || || ||2 2 2

m nU V

ij ij ij Fro Froi j

R R L Y U V I R R U V

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U

An Enhanced Model

• Local Similarity Calculation– Cosine Similarity

– Pearson Correlation Coefficient (PCC)

|| || || ||( , ) j k

j k

r rSim j k

r r

, ,( ) ( )

2 2, ,

( ) ( ) ( ) ( )

( ) ( )( , )

( ) ( )

j ku j u ku U i U j

j ku j u ku U j U k u U j U k

r r r rSim j k

r r r r

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An Enhanced Model

• Global And Local Consistent Similarity (GLCS)– Global similarity from item latent matrix

– Global and Local similarity combination

– Weighted undirected item-graph

( , ) ( , ) (1 ) ( , )j kGLCS j k sim v v sim j k

V( , ) cos ( , )j k j ksim v v ine v v

( , , )G V E W

( , )jkW GLCS j k

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An Enhanced Model

• Green’s Function Calculation (An Example)– Given an item-graph

– Calculate the Laplacian matrix L= D-W

1 2

43

5

0.2

0.25

0.40.6

0.50.1

0.8

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

W=

D=

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An Enhanced Model

• Green’s Function Calculation– Defined as the inverse of matrix L with zero-

mode discarded

* 1

2

1( )

Tni i

i i

v vG L

D W

,i i iLv v 1 20 ... n

1 0 without

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An Enhanced Model

• Label Propagation Recommendation– rating as label ;– Closed form label propagation:

1

1, argmax,

0,

l

k ji ikijk

k G yy l j n

otherwise

Label PropagationLabel data Unlabeled data

ijR jy

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Experimental Analysis

• Dataset– MovieLens dataset

• Metrics– Mean Absolute Error (MAE)– Mean Zero-one Error (MZOE)– Rooted Mean Squared Error (RMSE)

#Rating #Item #User #Rating Range

#Training Data

#Test Data

Sparsity Level

100,000 1682 943 1~5 80,000 20,000 6.3%

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Experimental Analysis

• Impact of Weight Parameter

k=10

k=5

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Experimental Analysis

• Performance Comparison– Previous Green’s function model (GCOS, GPCC),

[KDD ’07]

– Item-based recommendation (ICOS, IPCC)– User-based recommendation (UCOS, UPCC)

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Conclusion

• Latent features provide global similarity.• Global and local consistent similarity can

improve item-graph construction.• The enhanced model outperformed other

memory-based methods and previous model.

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Q&A

Thank you!

ICONIP 2010, Sydney, Australia 20

PMF

• Probabilistic Matrix Factorization– Define a conditional distribution over the

observed ratings as:

ICONIP 2010, Sydney, Australia 21

2 2

1 1

( | , , ) [ ( | , )]ijIm n

Tij i j

i j

p R U V N R U V

1, 0

0, 0ij

ijij

RI

R

Gaussian Distribution

PMF

• PMF– Assume zero-mean spherical Gaussian priors

on user and item feature

– By Bayesian Inference:

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2 2

1

2 2

1

( | ) ( | 0, )

( | ) ( | 0, )

m

U i Ui

n

V j Vj

p U N U I

p V N V I

2 2 2 2 2 2( , | , , , ) ( | , , ) ( | ) ( | )U V U Vp U V R p R U V p U p V

PMF

• PMF– Optimization: to maximize the log likelihood of

the posterior distribution:

– Using Gradient Decent in Y, U, V to get local optimal.

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2 2 200

1 1

1min || || min ( , , ) min ( ) || || || ||2 2 2

m nU V

ij ij ij Fro Froi j

R R L Y U V I R R U V

Algorithm

• Algorithm

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