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Multiverse Recommendation: N-dimensional Tensor
Factorization for Context-aware Collaborative
Filtering
Alexandros Karatzo...
Context in Recommender Systems
2
Context in Recommender Systems
Context is an important factor to consider in personalized
Recommendation
2
Context in Recommender Systems
Context is an important factor to consider in personalized
Recommendation
2
Context in Recommender Systems
Context is an important factor to consider in personalized
Recommendation
2
Context in Recommender Systems
Context is an important factor to consider in personalized
Recommendation
2
Current State of the Art in Context- aware
Recommendation
3
Current State of the Art in Context- aware
Recommendation
Pre-Filtering Techniques
3
Current State of the Art in Context- aware
Recommendation
Pre-Filtering Techniques
Post-Filtering Techniques
3
Current State of the Art in Context- aware
Recommendation
Pre-Filtering Techniques
Post-Filtering Techniques
Contextual mo...
Current State of the Art in Context- aware
Recommendation
Pre-Filtering Techniques
Post-Filtering Techniques
Contextual mo...
Collaborative Filtering problem setting
Typically data sizes e.g. Netlix data n = 5 × 105, m = 17 × 103
4
Standard Matrix Factorization
Find U ∈ Rn×d and M ∈ Rd×m so that F = UM
minimizeU,ML(F, Y) + λΩ(U, M)
Movies!
Users!
U!
M!...
Multiverse Recommendation: Tensors for Context
Aware Collaborative Filtering
Movies!
Users!
6
Tensors for Context Aware Collaborative Filtering
Movies!
Users!
U!
C!
M!
S!
7
Tensors for Context Aware Collaborative Filtering
Movies!
Users!
U!
C!
M!
S!
Fijk = S ×U Ui∗ ×M Mj∗ ×C Ck∗
R[U, M, C, S] :...
Regularization
Ω[F] = λM M 2
F + λU U 2
F + λC C 2
F
Ω[S] := λS S 2
F (1)
8
Squared Error Loss Function
Many implementations of MF used a simple squared error regression
loss function
l(f, y) =
1
2
...
Absolute Error Loss Function
Alternatively one can use the absolute error loss function
l(f, y) = |f − y|
thus the loss ov...
Optimization - Stochastic Gradient Descent for TF
The partial gradients with respect to U, M, C and S can then be
written ...
Optimization - Stochastic Gradient Descent for TF
The partial gradients with respect to U, M, C and S can then be
written ...
Optimization - Stochastic Gradient Descent for TF
Movies!
Users!
U!
C!
M!
S!
12
Optimization - Stochastic Gradient Descent for TF
Movies!
Users!
U!
C!
M!
S!
13
Optimization - Stochastic Gradient Descent for TF
Movies!
Users!
U!
C!
M!
S!
14
Optimization - Stochastic Gradient Descent for TF
Movies!
Users!
U!
C!
M!
S!
15
Optimization - Stochastic Gradient Descent for TF
Movies!
Users!
U!
C!
M!
S!
16
Experimental evaluation
We evaluate our model on contextual rating data and computing the
Mean Absolute Error (MAE),using ...
Data
Data set Users Movies Context Dim. Ratings Scale
Yahoo! 7642 11915 2 221K 1-5
Adom. 84 192 5 1464 1-13
Food 212 20 2 ...
Context Aware Methods
Pre-filtering based approach, (G. Adomavicius et.al), computes
recommendations using only the ratings...
Results: Context vs. No Context
No
context
Tensor
Factorization
1.9
2.0
2.1
2.2
2.3
2.4
2.5
MAE
(a)
No
context
Tensor
Fact...
Yahoo Artificial Data
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
MAE
α=0.1
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
α...
Yahoo Artificial Data
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Probability of contextual influence
0.60
0.65
0.70
0.75
0.80
...
Tensor Factorization
Reduction Item-Split Tensor
Factorization
1.9
2.0
2.1
2.2
2.3
2.4
2.5
MAE
Figure: Comparison of conte...
Tensor Factorization
No
context
Reduction Tensor
Factorization
0.80
0.85
0.90
0.95
MAE
Figure: Comparison of context-aware...
Conclusions
Tensor Factorization methods seem to be promising for CARS
Many different TF methods exist
Future work: extend...
Thank You !
26
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Multiverse Recommendation: N-dimensional Tensor Factorization for Context-aware Collaborative Filtering

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Slides from RecSys 2010 presentation.
Context has been recognized as an important factor to con- sider in personalized Recommender Systems. However, most model-based Collaborative Filtering approaches such as Ma- trix Factorization do not provide a straightforward way of integrating context information into the model. In this work, we introduce a Collaborative Filtering method based on Tensor Factorization, a generalization of Matrix Factoriza- tion that allows for a flexible and generic integration of con- textual information by modeling the data as a User-Item- Context N-dimensional tensor instead of the traditional 2D User-Item matrix. In the proposed model, called Multiverse Recommendation, different types of context are considered as additional dimensions in the representation of the data as a tensor

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Multiverse Recommendation: N-dimensional Tensor Factorization for Context-aware Collaborative Filtering

  1. 1. Multiverse Recommendation: N-dimensional Tensor Factorization for Context-aware Collaborative Filtering Alexandros Karatzoglou1 Xavier Amatriain 1 Linas Baltrunas 2 Nuria Oliver 2 1Telefonica Research Barcelona, Spain 2Free University of Bolzano Bolzano, Italy November 4, 2010 1
  2. 2. Context in Recommender Systems 2
  3. 3. Context in Recommender Systems Context is an important factor to consider in personalized Recommendation 2
  4. 4. Context in Recommender Systems Context is an important factor to consider in personalized Recommendation 2
  5. 5. Context in Recommender Systems Context is an important factor to consider in personalized Recommendation 2
  6. 6. Context in Recommender Systems Context is an important factor to consider in personalized Recommendation 2
  7. 7. Current State of the Art in Context- aware Recommendation 3
  8. 8. Current State of the Art in Context- aware Recommendation Pre-Filtering Techniques 3
  9. 9. Current State of the Art in Context- aware Recommendation Pre-Filtering Techniques Post-Filtering Techniques 3
  10. 10. Current State of the Art in Context- aware Recommendation Pre-Filtering Techniques Post-Filtering Techniques Contextual modeling 3
  11. 11. Current State of the Art in Context- aware Recommendation Pre-Filtering Techniques Post-Filtering Techniques Contextual modeling The approach presented here fits in the Contextual Modeling category 3
  12. 12. Collaborative Filtering problem setting Typically data sizes e.g. Netlix data n = 5 × 105, m = 17 × 103 4
  13. 13. Standard Matrix Factorization Find U ∈ Rn×d and M ∈ Rd×m so that F = UM minimizeU,ML(F, Y) + λΩ(U, M) Movies! Users! U! M! 5
  14. 14. Multiverse Recommendation: Tensors for Context Aware Collaborative Filtering Movies! Users! 6
  15. 15. Tensors for Context Aware Collaborative Filtering Movies! Users! U! C! M! S! 7
  16. 16. Tensors for Context Aware Collaborative Filtering Movies! Users! U! C! M! S! Fijk = S ×U Ui∗ ×M Mj∗ ×C Ck∗ R[U, M, C, S] := L(F, Y) + Ω[U, M, C] + Ω[S] 7
  17. 17. Regularization Ω[F] = λM M 2 F + λU U 2 F + λC C 2 F Ω[S] := λS S 2 F (1) 8
  18. 18. Squared Error Loss Function Many implementations of MF used a simple squared error regression loss function l(f, y) = 1 2 (f − y)2 thus the loss over all users and items is: L(F, Y) = n i m j l(fij, yij) Note that this loss provides an estimate of the conditional mean 9
  19. 19. Absolute Error Loss Function Alternatively one can use the absolute error loss function l(f, y) = |f − y| thus the loss over all users and items is: L(F, Y) = n i m j l(fij, yij) which provides an estimate of the conditional median 10
  20. 20. Optimization - Stochastic Gradient Descent for TF The partial gradients with respect to U, M, C and S can then be written as: ∂Ui∗ l(Fijk , Yijk ) = ∂Fijk l(Fijk , Yijk )S ×M Mj∗ ×C Ck∗ ∂Mj∗ l(Fijk , Yijk ) = ∂Fijk l(Fijk , Yijk )S ×U Ui∗ ×C Ck∗ ∂Ck∗ l(Fijk , Yijk ) = ∂Fijk l(Fijk , Yijk )S ×U Ui∗ ×M Mj∗ ∂Sl(Fijk , Yijk ) = ∂Fijk l(Fijk , Yijk )Ui∗ ⊗ Mj∗ ⊗ Ck∗ 11
  21. 21. Optimization - Stochastic Gradient Descent for TF The partial gradients with respect to U, M, C and S can then be written as: ∂Ui∗ l(Fijk , Yijk ) = ∂Fijk l(Fijk , Yijk )S ×M Mj∗ ×C Ck∗ ∂Mj∗ l(Fijk , Yijk ) = ∂Fijk l(Fijk , Yijk )S ×U Ui∗ ×C Ck∗ ∂Ck∗ l(Fijk , Yijk ) = ∂Fijk l(Fijk , Yijk )S ×U Ui∗ ×M Mj∗ ∂Sl(Fijk , Yijk ) = ∂Fijk l(Fijk , Yijk )Ui∗ ⊗ Mj∗ ⊗ Ck∗ We then iteratively update the parameter matrices and tensors using the following update rules: Ut+1 i∗ = Ut i∗ − η∂UL − ηλUUi∗ Mt+1 j∗ = Mt j∗ − η∂ML − ηλMMj∗ Ct+1 k∗ = Ct k∗ − η∂CL − ηλCCk∗ St+1 = St − η∂Sl(Fijk , Yijk ) − ηλSS where η is the learning rate. 11
  22. 22. Optimization - Stochastic Gradient Descent for TF Movies! Users! U! C! M! S! 12
  23. 23. Optimization - Stochastic Gradient Descent for TF Movies! Users! U! C! M! S! 13
  24. 24. Optimization - Stochastic Gradient Descent for TF Movies! Users! U! C! M! S! 14
  25. 25. Optimization - Stochastic Gradient Descent for TF Movies! Users! U! C! M! S! 15
  26. 26. Optimization - Stochastic Gradient Descent for TF Movies! Users! U! C! M! S! 16
  27. 27. Experimental evaluation We evaluate our model on contextual rating data and computing the Mean Absolute Error (MAE),using 5-fold cross validation defined as follows: MAE = 1 K n,m,c ijk Dijk |Yijk − Fijk | 17
  28. 28. Data Data set Users Movies Context Dim. Ratings Scale Yahoo! 7642 11915 2 221K 1-5 Adom. 84 192 5 1464 1-13 Food 212 20 2 6360 1-5 Table: Data set statistics 18
  29. 29. Context Aware Methods Pre-filtering based approach, (G. Adomavicius et.al), computes recommendations using only the ratings made in the same context as the target one Item splitting method (L. Baltrunas, F. Ricci) which identifies items which have significant differences in their rating under different context situations. 19
  30. 30. Results: Context vs. No Context No context Tensor Factorization 1.9 2.0 2.1 2.2 2.3 2.4 2.5 MAE (a) No context Tensor Factorization 0.80 0.85 0.90 0.95 MAE (b) Figure: Comparison of matrix (no context) and tensor (context) factorization on the Adom and Food data. 20
  31. 31. Yahoo Artificial Data 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 MAE α=0.1 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 α=0.5 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 α=0.9 No Context Reduction Item-Split Tensor Factorization Figure: Comparison of context-aware methods on the Yahoo! artificial data 21
  32. 32. Yahoo Artificial Data 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probability of contextual influence 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95MAE No Context Reduction Item-Split Tensor Factorization 22
  33. 33. Tensor Factorization Reduction Item-Split Tensor Factorization 1.9 2.0 2.1 2.2 2.3 2.4 2.5 MAE Figure: Comparison of context-aware methods on the Adom data. 23
  34. 34. Tensor Factorization No context Reduction Tensor Factorization 0.80 0.85 0.90 0.95 MAE Figure: Comparison of context-aware methods on the Food data. 24
  35. 35. Conclusions Tensor Factorization methods seem to be promising for CARS Many different TF methods exist Future work: extend to implicit taste data Tensor representation of context data seems promising 25
  36. 36. Thank You ! 26

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