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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
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 modeling
3
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
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!
5
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] := L(F, Y) + Ω[U, M, C] + Ω[S]
7
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
(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
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
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
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
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 5-fold cross validation defined as
follows:
MAE =
1
K
n,m,c
ijk
Dijk |Yijk − Fijk |
17
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
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
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
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
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
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
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
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
Thank You !
26

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

  • 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
  • 3. Context in Recommender Systems Context is an important factor to consider in personalized Recommendation 2
  • 4. Context in Recommender Systems Context is an important factor to consider in personalized Recommendation 2
  • 5. Context in Recommender Systems Context is an important factor to consider in personalized Recommendation 2
  • 6. Context in Recommender Systems Context is an important factor to consider in personalized Recommendation 2
  • 7. Current State of the Art in Context- aware Recommendation 3
  • 8. Current State of the Art in Context- aware Recommendation Pre-Filtering Techniques 3
  • 9. Current State of the Art in Context- aware Recommendation Pre-Filtering Techniques Post-Filtering Techniques 3
  • 10. Current State of the Art in Context- aware Recommendation Pre-Filtering Techniques Post-Filtering Techniques Contextual modeling 3
  • 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. Collaborative Filtering problem setting Typically data sizes e.g. Netlix data n = 5 × 105, m = 17 × 103 4
  • 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. Multiverse Recommendation: Tensors for Context Aware Collaborative Filtering Movies! Users! 6
  • 15. Tensors for Context Aware Collaborative Filtering Movies! Users! U! C! M! S! 7
  • 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. Regularization Ω[F] = λM M 2 F + λU U 2 F + λC C 2 F Ω[S] := λS S 2 F (1) 8
  • 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. 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. 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. 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. Optimization - Stochastic Gradient Descent for TF Movies! Users! U! C! M! S! 12
  • 23. Optimization - Stochastic Gradient Descent for TF Movies! Users! U! C! M! S! 13
  • 24. Optimization - Stochastic Gradient Descent for TF Movies! Users! U! C! M! S! 14
  • 25. Optimization - Stochastic Gradient Descent for TF Movies! Users! U! C! M! S! 15
  • 26. Optimization - Stochastic Gradient Descent for TF Movies! Users! U! C! M! S! 16
  • 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. 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. 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. 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. 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. 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. 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
  • 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