SLIM : Sparse Linear Methods for Top-N Recommender Systems Xia Ning and George Karypis Computer Science & Engineering University of Minnesota, Minneapolis, MN Email: {xning,karypis@cs.umn.edu} December 14, 2011
Introduction Methods Materials Experimental Results Conclusions 2/25 Outline Introduction 1 Top-N Recommender Systems Definitions and Notations The State-of-the-Art methods Methods 2 Sparse LInear Methods for top-N Recommendation Learning W for SLIM SLIM with Feature Selection Materials 3 Experimental Results 4 SLIM on Binary Data Top-N Recommendation Performance SLIM for Long-Tail Distribution SLIM Regularization Effects SLIM on Rating Data Conclusions 5 Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 3/25 Outline Introduction 1 Top-N Recommender Systems Definitions and Notations The State-of-the-Art methods Methods 2 Sparse LInear Methods for top-N Recommendation Learning W for SLIM SLIM with Feature Selection Materials 3 Experimental Results 4 SLIM on Binary Data Top-N Recommendation Performance SLIM for Long-Tail Distribution SLIM Regularization Effects SLIM on Rating Data Conclusions 5 Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 4/25 Top-N Recommender Systems ❑ Top-N recommendation ❑ E-commerce: huge amounts of products ❑ Recommend a short ranked list of items for users ❑ Top-N recommender systems ❑ Neighborhood-based Collaborative Filtering ( CF ) ❑ Item based [2]: fast to generate recommendations, low recommendation quality ❑ Model-based methods [1, 3, 5] ❑ Matrix Factorization ( MF ) models: slow to learn the models, high recommendation quality ❑ SLIM : Sparse LInear Methods ❑ Fast and high recommendation quality Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 5/25 Definitions and Notations Table 1: Definitions and Notations Def Descriptions user u i t j item all users ( |U| = n ) U all items ( |T | = m ) T user-item purchase/rating matrix, size n × m A W item-item similarity matrix/coefficient matrix a T The i -th row of A , the purchase/rating history of u i on T i The j -th column of A , the purchase/rating history of U on t j a j ❑ Row vectors are represented by having the transpose supscript T , otherwise by default they are column vectors. ❑ Use matrix/vector notations instead of user/item purchase/rating profiles Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 6/25 The State-of-the-Art Methods Item-based Collaborative Filtering (1) ❑ Item-based k -nearest-neighbor ( itemkNN ) CF ❑ Identify a set of similar items ❑ Item-item similarity: ❑ Calculated from A ❑ Cosine similarity measure 2nd nn 1st nn t 1 t 2 t 3 . . . . . . t j . . . . . . t 1 t 2 t 3 . . . . . . t j . . . . . . t m − 1 t m t m − 1 t m s s s u 1 1 t 1 s s s u 2 1 1 1 t 2 s s s u 3 1 1 1 t 3 . . . . . . . . . . . . s s s 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . s u i t j 1 . . . . . . s 1 . . . . . . s s s 1 1 1 s u n − 1 1 t m − 1 s s u n 1 1 t m A W Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 7/25 The State-of-the-Art Methods Item-based Collaborative Filtering (2) t 1 t 2 t 3 . . . . . . t j . . . . . . u T u T t m − 1 t m ∗· ∗· p s s s t 1 t 1 t 1 s s s t 2 t 2 t 2 p s s s t 3 t 3 1 t 3 . . . . . . . . . . . . . . . . . . p × = s s s 1 . . . . . . . . . . . . p s t j t j t j . . . . . . . . . s . . . p . . . . . . s s s 1 p s t m − 1 t m − 1 t m − 1 s s t m t m 1 t m ❑ itemkNN recommendation ❑ Recommend similar items to what the user has purchased a T i = a T ˜ i × W ❑ Fast: sparse item neighborhood ❑ Low quality: no knowledge is learned Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 8/25 The State-of-the-Art Methods Matrix Factorization (1) ❑ Latent factor models ❑ Factorize A into low-rank user factors ( U ) and item factors ( V T ) ❑ U and V T represent user and item characteristics in a common latent space ❑ Formulated as an optimization problem 1 F + β F + λ 2 � A − UV T � 2 2 � U � 2 2 � V T � 2 minimize F U , V T t 1 t 2 t 3 . . . . . . t j . . . . . . l 1 l 2 . . . l k t m − 1 t m u u u u u 1 1 u 1 u u u u u 2 u 2 1 1 1 t 1 t 2 t 3 . . . . . . t j . . . . . . u u u u u 3 1 1 1 u 3 t m − 1 t m . . . . . . u u u u v v v v v v v v v v l 1 . . . . . . u u u u v v v v v v v v v v 1 1 1 l 2 × . . . . . . . . . . . . u u u u v v v v v v v v v v u i u i . . . 1 . . . . . . u u u u v v v v v v v v v v 1 l k . . . . . . u u u u 1 1 1 u u u u u n − 1 1 u k − 1 u n u k u u u u 1 1 × V T A U Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 9/25 The State-of-the-Art Methods Matrix Factorization (2) u T ∗· p t 1 p t 2 t 1 t 2 t 3 . . . . . . t j . . . . . . p t 3 t m − 1 t m . . . p v v v v v v v v v v l 1 l 2 . . . l k l 1 . . . p u u u u × v v v v v v v v v v = u ∗ l 2 p v v v v v v v v v v t j . . . . . . p v v v v v v v v v v l k . . . p p t m − 1 p t m ❑ MF recommendation ❑ Prediction: dot product in the latent space a ij = U T ˜ i · V j ❑ Slow: dense U and V T ❑ High quality: user tastes and item properties are learned Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 10/25 Outline Introduction 1 Top-N Recommender Systems Definitions and Notations The State-of-the-Art methods Methods 2 Sparse LInear Methods for top-N Recommendation Learning W for SLIM SLIM with Feature Selection Materials 3 Experimental Results 4 SLIM on Binary Data Top-N Recommendation Performance SLIM for Long-Tail Distribution SLIM Regularization Effects SLIM on Rating Data Conclusions 5 Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 11/25 SLIM for top-N Recommendation ❑ Motivations: ❑ recommendations generated fast ❑ high-quality recommendations ❑ “have my cake and eat it too” ❑ Key ideas: ❑ retain the nature of itemkNN : sparse W ❑ optimize the recommendation performance: learn W from A ❑ sparsity structures ❑ coefficient values Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 12/25 Learning W for SLIM ❑ The optimization problem: F + β 1 2 � A − AW � 2 2 � W � 2 minimize F + λ � W � 1 W (1) subject to W ≥ 0 diag ( W ) = 0 , Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 12/25 Learning W for SLIM ❑ The optimization problem: F + β 1 2 � A − AW � 2 2 � W � 2 minimize F + λ � W � 1 W (1) subject to W ≥ 0 diag ( W ) = 0 , ❑ Computing W : ❑ The columns of W are independent: easy to parallelize ❑ The decoupled problems: 1 2 + β 2 � a j − A w j � 2 2 � w j � 2 minimize 2 + λ � w j � 1 w j (2) subject to w j ≥ 0 w j , j = 0 , Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 13/25 Reducing model learning time 1 2 + β 2 � a j − A w j � 2 2 � w j � 2 minimize 2 + λ � w j � 1 w j ❑ fsSLIM : SLIM with f eature s election ❑ Prescribe the potential non-zero structure of w j ❑ Select a subset of columns from A ❑ itemkNN item-item similarity matrix a j u 1 1 1 1 1 1 u 1 u 2 1 1 u 2 1 1 1 u 3 1 1 1 1 1 u 3 1 . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . u i u j 1 . . . . . . 1 1 . . . . . . 1 1 1 1 1 1 1 1 u n − 1 1 1 u m − 1 1 u n 1 1 u m 1 1 A ′ A 1 2 + β 2 � a j − A ′ w j � 2 2 � w j � 2 minimize 2 + λ � w j � 1 w j Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
Introduction Methods Materials Experimental Results Conclusions 14/25 Outline Introduction 1 Top-N Recommender Systems Definitions and Notations The State-of-the-Art methods Methods 2 Sparse LInear Methods for top-N Recommendation Learning W for SLIM SLIM with Feature Selection Materials 3 Experimental Results 4 SLIM on Binary Data Top-N Recommendation Performance SLIM for Long-Tail Distribution SLIM Regularization Effects SLIM on Rating Data Conclusions 5 Xia Ning and George Karypis SLIM : Sparse Linear Methodsfor Top-N Recommender Systems •
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