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Collaborative filtering using orthogonal nonnegative matrix

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  • 1. COLLABORATIVE FILTERING USING ORTHOGONAL NONNEGATIVE MATRIX Presenter: Meng-Lun Wu Authors: Gang Chen, Fei Wang and Changshui Zhang Source: Information Processing and Management (2009), pp. 368-379 1
  • 2. OUTLINE  Introduction  Related Work  Orthogonal nonnegative matrix tri-factorization  Framework  Experiments  Conclusion 2
  • 3. INTRODUCTION  Collaborative filtering can predict a test user’s rating for new items based on similar users.  Collaborative filtering can be categorized into…  Memory-based (similarity)  user-based and item-based  Model-based  Establish a model using training examples. 3
  • 4. INTRODUCTION (CONT.)  This paper apply orthogonal nonnegative matrix tri-factorization(ONMTF) to circumvent the two kinds of collaborative filtering.  ONMTF is applied to simultaneously co-cluster the rows and columns, and attain individual predictions for an unknown test rating.  This paper possesses the following superiorities:  Sparsity problem  Scalability problem  Fusing prediction results 4
  • 5. RELATED WORK  Researchers have proposed some hybrid approaches in order to combine the memory based and model based approaches.  Xue et. al. (2005) resolve the sparsity and scalability by using clusters to smooth ratings and clustering.  However, Xue only consider user-based approach, this paper extend the idea to integrate model based, user-based and item-based approaches. 5
  • 6. RELATED WORK (CONT.)  Matrix decomposition can used to solve the co- clustering problem.  Ding et al. (2005) proposed co-clustering based on nonnegative matrix factorization. (NMF)  In 2006, they proposed ONMTF.  Long et al. (2005) provided co-clustering by block value decomposition. 6
  • 7. ORTHOGONAL NONNEGATIVE MATRIX TRI-FACTORIZATION  The NMF is first brought into machine learning and data mining fields by Lee et al. (2001).  Ding et al. (2006) proved the equivalence between NMF and K-means, and extended NMF to ONMTF.  The idea is to approximate the original matrix X to the combination matrix, and the optimization problem is 7 lnlkkpnp TTT VSU andwhere IVVIUUtsUSVX × + × + × + × + ≥≥≥ ℜ∈ℜ∈ℜ∈ℜ∈ ==− VS,U,X ,..,min 2 0,0,0
  • 8. ORTHOGONAL NONNEGATIVE MATRIX TRI-FACTORIZATION (CONT.)  The optimization problem can be solved using the following update rules.  After co-clustering, we could get the user centroid SVT and item centroid US. 8 ( ) ( ) ( )ik TT ik T ikik ik TT ik T ikik jk TT jk T jkjk VUSVU XVU SS XVSUU XVS UU USXVV USX VV )( )( )( ← ← ←
  • 9. FRAMEWORK  Notations  X = [u1,…,up]T , uj=(xj1,…,xjn)T , j∈{1,…,p}  X = [i1,…,in], im=(x1m,…,xpm)T , m∈{1,…,n} 9
  • 10. MEMORY-BASED APPROACHES  User’s neighbor selection  Compute the similarities between a user and all the user-cluster centroids SVT .  Select the top K user cluster as the user set uh.  The item’s neighbor selection is similar.  The cosine similarity between the j1th user and the j2th user.  Given an user-item pair <uj, im>, where uh∈{the most similar K-user of uj}. 10 ∑∑ ∑ == = = n m mj n m mj n m mjmj xx xx sim 1 2 1 2 1 )()( ))(( ),( 21 21 21 jj uu ∑ ∑ − += h h u jh u jh uu uu ),( ))(,( sim uusim ux hhm jjm
  • 11. MEMORY-BASED APPROACHES  The adjusted-cosine similarity between the m1th and m2th items. (T is the set of users who both rated m1 and m2)  Given an user-item pair <uj, im>, where ih∈{the most similar K-items of im}  The final prediction result could be linearly combined the three different types of predictions. 11 ∑∑ ∑ ∈∈ ∈ −− −− = Tt ttmTt ttm Tt ttmttm uxux uxux sim 22 )()( ))(( ),( 21 21 21 mm ii ∑ ∑= h h i mh i mh ii ii ),( ))(,( sim xsim x jh jm jmjmjmjm xixuxnx ~~ )1)(1(~~)1(~~~ λδλδλ −−+−+=
  • 12. ALGORITHM 1. The user-item matrix X is factorized as USVT by using ONMTF. 2. Calculate the similarities between the test user/item and user/item-cluster centroids. 3. Sort the similarities and select the most similar C user/item clusters as the test user/item neighbor candidate set. 4. Identify the most K neighbors of the test user/item by searching for the user/item candidate set. 5. Predict the unknown ratings by using user based and item based approaches. 6. Linearly combine three different predictions. 12
  • 13. EXPERIMENTS  Dataset  MovieLens: 500 users and 1000 items (1-5 scales)  Training set: the first 100, 200 and 300 users, called ML_100, ML_200 and ML_300.  Testing set: the last 200 users  We randomly selected 5, 10 and 20 items rated by test users, called Given5, Given10 and Given20.  Evaluation metric  Mean absolute error (MAE) as evaluation metric.  Where N is the number of tested ratings. 13 N xx MAE mj jmjm∑ − = , ~
  • 14. DIFFERENT CLUSTER 14  The ML_300 dataset is used for training, and try 10 different values of k or l (2,5,10,20,…,80)
  • 15. PERCENTAGE OF NEIGHBORS  The percentage of pre-selected neighbors reaches around 30%. 15
  • 16. SIZE OF NEIGHBORS 16
  • 17. COMBINATION COEFFICIENTS  Fix λ=0, the optimal value of δ is approximately between 0.5 and 0.7. 17
  • 18. COMBINATION COEFFICIENTS (CONT.)  Fix δ=0.6, the optimal value of λ is approximately between 0.2 and 0.4. 18
  • 19. PERFORMANCE COMPARISON 19  Wang et al., 2006, similarity fusion (SF2)  Xue et al., 2005, cluster-based Pearson correlation coefficient (SCBPCC)  Rennie and Srebro, 2005, maximum margin matrix factorization (MMMF)  Ungar and Foster, 1999, cluster-based collaborative filtering (CBCF)  Hofmann and Puzicha, 1999, aspect model (AM)  Pennock et al., 2000, personality diagnosis (PD)  Breese et al., 1998, user-based Pearson correlation coefficient (PCC)
  • 20. CONCLUSIONS  This paper presented a novel fusion framework for collaborative filtering.  The model-based and memory-based and naturally assembled via ONMTF.  Empirical studies verified our framework effectively improves the prediction accuracy.  Future work is investigate new co-clustering techniques and develop better fusion models. 20

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