MF-DMPC: Matrix Factorization with Dual Multiclass Preference Context for Rating Prediction Weike Pan Zhong Ming Jing Lin National Engineering Laboratory for Big Data System Computing Technology, College of Computer Science and Software Engineering, Shenzhen University, China linjing4@email.szu.edu.cn, { panweike,mingz } @szu.edu.cn Lin, Pan and Ming (SZU) MF-DMPC 1 / 26
Introduction Context in RS External Context La La Land Comedy, Drama, Music … ... Avatar Action, Adventure, Fantasy Item … ... content Weather Internal Context items … m-1 m 1 2 3 Social users network 1 ? 4 ? ? ? Date 2 3 ? 1 ? ? 3 5 ? ? ? 2 … Place n-1 ? ? ? 3 4 n 2 1 ? ? 5 …… Rating matrix Using external context in RS may cause the problems of model inflexibility, computational burden and systems incompatibility. So algorithms that only use internal context (collaborative filtering) still occupy an important place in the research community. Lin, Pan and Ming (SZU) MF-DMPC 2 / 26
Introduction Motivation MF-MPC combines the two complementary categories of collaborative filtering – neighborhood-based and model-based. MF-MPC is an improved method of SVD (a typical kind of matrix factorization method) by adding a matrix transformed from the multiclass preference context (MPC) of a certain user, which represents the user similarities in a neighborhood-based method. In this paper, we further introduce a matrix factorization model that combines not only user similarities but also item similarities. Lin, Pan and Ming (SZU) MF-DMPC 3 / 26
Introduction The Derivation Process of MF-DMPC (1) user-specific latent feature vector item-specific latent feature vector user-specific latent preference vector user bias,item bias, , global average (a) ) user-based MF-MPC PC model (u u = = 1,2, … ,n; ; i i = = 1,2, … ,m) According to Pan et al., MF-MPC refers to matrix factorization with (user-based) multiclass preference context. Lin, Pan and Ming (SZU) MF-DMPC 4 / 26
Introduction The Derivation Process of MF-DMPC (2) item-specific latent preference vector user-specific latent feature vector item-specific latent feature vector user bias,item bias, , global average (b) ) item-based MF-MPC PC model (u u = = 1,2, … ,n; ; i i = = 1,2, … ,m) Symmetrically, we introduce item-based multiclass preference context to represent item similarities. Lin, Pan and Ming (SZU) MF-DMPC 5 / 26
Introduction The Derivation Process of MF-DMPC (3) item-specific latent preference vector user-specific latent feature vector item-specific latent feature vector user-specific latent preference vector user bias,item bias, , global average (c) ) MF MF-DMPC PC model (u u = = 1,2, … ,n; ; i i = = 1,2, … ,m) Finally, we introduce both user-based and item-based MPC (dual MPC) into the prediction rule to obtain our improved model called matrix factorization with dual multi-class preference context (MF-DMPC). Lin, Pan and Ming (SZU) MF-DMPC 6 / 26
Introduction Advantage of Our Solution MF-DMPC inherits high accuracy and good explainability of MF-MPC and performs even better. As a matter of fact, our model is a more generic method which successfully exploits the complementarity between user-based and item-based neighborhood information. Lin, Pan and Ming (SZU) MF-DMPC 7 / 26
Preliminaries Problem Definition In this paper, we study the problem of making good use of internal context in recommendation systems. We will only need an incomplete rating matrix R = { ( u , i , r ui ) } for our task, where u represents one of the ID numbers of n users (or rows), i represents one of the ID numbers of m items (or columns), and r ui is the recorded rating of user u to item i . As a result, we will build an improved model to estimate the missing entries of the rating matrix. Lin, Pan and Ming (SZU) MF-DMPC 8 / 26
Preliminaries Notations Table: Some notations and explanations. Symbol Meaning user number n item number m u , u ′ user ID i , i ′ item ID multiclass preference set M r ui ∈ M rating of user u to item i R = { ( u , i , r ui ) } rating records of training data y ui ∈ { 0 , 1 } indicator, y ui = 1 if ( u , i , r ui ) ∈ R and y ui = 0 otherwise I r u , r ∈ M items rated by user u with rating r I u items rated by user u U r i , r ∈ M users who rate item i with rating r U i users who rate item i µ ∈ R global average rating value b u ∈ R user bias b i ∈ R item bias d ∈ R number of latent dimensions U u · , N r u · ∈ R 1 × d user-specific latent feature vector V i · , M r i · ∈ R 1 × d item-specific latent feature vector R te = { ( u , i , r ui ) } rating records of test data ˆ r ui predicted rating of user u to item i T iteration number in the algorithm Lin, Pan and Ming (SZU) MF-DMPC 9 / 26
Preliminaries Prediction Rule of SVD In the state-of-the-art matrix factorization based model – SVD model, the prediction rule for the rating of user u to item i is as follows, r ui = U u · V T ˆ i · + b u + b i + µ, (1) where U u · ∈ R 1 × d and V i · ∈ R 1 × d are the user-specific and item-specific latent feature vectors, respectively, and b u , b i and µ are the user bias, the item bias and the global average, respectively. Lin, Pan and Ming (SZU) MF-DMPC 10 / 26
Preliminaries Preference Generalization Probability of MF-MPC In the MF-MPC model, the rating of user u to item i , r ui , can be represented in a probabilistic way as, P ( r ui | ( u , i ); ( u , i ′ , r ui ′ ) , i ′ ∈ ∪ r ∈ M I r u \{ i } ) , (2) which means that r ui is dependent on not only the (user, item) pair ( u , i ), but also the examined items i ′ ∈ I u \{ i } and the categorical score r ui ′ ∈ M of each item. Notice that multiclass preference context (MPC) refers to the condition ( u , i ′ , r ui ′ ) , i ′ ∈ ∪ r ∈ M I r u \{ i } . Lin, Pan and Ming (SZU) MF-DMPC 11 / 26
Preliminaries Prediction Rule of MF-MPC In order to introduce MPC into MF based model, we need a user-specific latent preference vector ¯ U MPC for user u , u · 1 ¯ U MPC � � M r = i ′ · . (3) u · � |I r u \{ i }| r ∈ M i ′ ∈I r u \{ i } i · ∈ R 1 × d is a classified item-specific latent feature vector and Notice that M r 1 √ u \{ i }| plays as a normalization term for the preference of class r . |I r By adding the neighborhood information ¯ U MPC to SVD model, we get the u · MF-MPC prediction rule for the rating of user u to item i , r ui = U u · V T i · + ¯ U MPC V T ˆ i · + b u + b i + µ, (4) u · where U u · and V i · , b u , b i and µ are exactly the same with that of the SVD model. MF-MPC generates better recommendation performance than SVD and SVD++, and also contains them as particular cases. Lin, Pan and Ming (SZU) MF-DMPC 12 / 26
Our Model Dual Multiclass Preference Context We first define item-based multiclass preference context (item-based MPC) ¯ V MPC to represent item similarities. Symmetrically, we have i · 1 ¯ � � V MPC N r = u ′ · , (5) i · � |U r i \{ u }| r ∈ M u ′ ∈U r i \{ u } u · ∈ R 1 × d is a classified user-specific latent feature vector. where N r So we have the item-based MF-MPC prediction rule, i · + ¯ r ui = U u · V T V MPC U T ˆ u · + b u + b i + µ. (6) i · We can introduce both user-based and item-base neighborhood information into matrix factorization method by keeping both ¯ U MPC u · and ¯ V MPC in the model, collectively called dual multiclass preference i · context (DMPC). Lin, Pan and Ming (SZU) MF-DMPC 13 / 26
Our Model Prediction Rule of MF-DMPC For matrix factorization with dual multiclass preference context, the prediction rule for the rating of user u to item i is defined as follows, i · + ¯ i · + ¯ r ui = U u · V T U MPC V T V MPC U T ˆ u · + b u + b i + µ, (7) u · i · with all notations described above. Finally, we call our new model “MF-DMPC” in short. Lin, Pan and Ming (SZU) MF-DMPC 14 / 26
Our Model Optimization Problem With the prediction rule, we can learn the model parameters in the following minimization problem, n m y ui [1 r ri ) 2 + reg ( u , i )] , � � min 2( r ui − ˆ (8) Θ u =1 i =1 where reg ( u , i ) = α m i ′ || 2 F + α n u ′ || 2 u \{ i } || M r i \{ u } || N r � � � � F + i ′ ∈I r u ′ ∈U r 2 r ∈ M 2 r ∈ M 2 || U u · || 2 + α v 2 || V i · || 2 + β u 2 || b u · || 2 + β v 2 || b i · || 2 is the regularization term α u used to avoid overfitting, and Θ = { U u · , V i · , b u , b i , µ, M r i · , N r u · } , u = 1 , 2 , . . . , n , i = 1 , 2 , . . . , m , r ∈ M . Notice that the objective function of MF-DMPC is quite similar to that of MF-MPC. The difference lies in the “dual” MPC, i.e., ¯ V MPC U T u · in the i · prediction rule, and α n i \{ u } || N r u ′ || 2 � � F in the regularization u ′ ∈U r 2 r ∈ M term. Lin, Pan and Ming (SZU) MF-DMPC 15 / 26
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