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Factored Proximity Models for Top-NRecommendationsAthanasios N. Nikolakopoulos1, Vassilis Kalantzis2, EfstratiosGallopoulos3 and John D. Garofalakis3
1Digital Technology Center, University of Minnesota, USA.2Dept. of Computer Science & Engineering, University of Minnesota, USA.3Dept. of Computer Engineering & Informatics, University of Patras, Greece.August 08, 2017
IEEE International Conference on Big Knowledge, IEEE ICBK 2017
Introduction & Motivation i
Recommender Systems• Widely Applicable
Technology• Value for Customers• Value for Companies
Collaborative Filtering• Model: Ratings!• Recommendation task
• Rating Prediction• Top-N Lists
1
Introduction & Motivation ii
Sparsity Problem• Limits the quality recommendations;
especially for Long-Tail Items.• Intrinsic RS Characteristic
• Cold-Start Problem
Promising Approaches• Graph-based Models
(+) Good Performance(-) Scalability Issues
• Latent-Factor Models.Our Focus: Efficient and High-QualityTop-N Recommendations Even underExtreme Sparsity
Figure 1: Graph-Based Idea
Figure 2: Latent-Factor Idea2
EigenRec Framework
EigenRec Framework
EigenRec• Build a symmetric m×m Inter-Item Proximity Matrix A, each
element of which is defined to be a product of a Scaling and aSimilarity component.
[A]ij ≜ ξ(i, j) · κ(i, j)
• Build a Lower Dimensional Model using the principal eigenvectorsof A
• Project the Users’s feedback vectors onto the Latent Subspace:
Π ≜ RVfV⊺f
Simple Baseline Choices
Scaling function Similarity functions
ξ(i, j) ≜ f(i, j; d) = (∥ri∥∥rj∥)d κ(i, j) ≜
cos(vi, vj)
pc(vi, vj)
jaccard(vi, vj) 3
PureSVD within EigenRec
PureSVD
ΠPureSVD ≜ UfΣfQ⊺f ≡ · · · ≡ RQfQ
⊺f
Where Qf , the matrix containing the f principal eigenvectors of:
R⊺R =
users
items – r⊺vi–
×
items|
users rvj
|
=
items
items ·
∥rvi∥∥rvj
∥︸ ︷︷ ︸scaling
·cos θij︸ ︷︷ ︸similarity
,
• PureSVD ≡ EigenRec with Cosine similarity and f(i, j; 1)
4
Computing EigenRec
EigenRec:Input: Inter-Item proximity matrix A ∈ Rm×m. RatingMatrix R ∈ Rn×m. Latent Factors f .Output: Matrix Π ∈ Rn×m whose rows are the recom-mendation vectors for every user.1: qj = 0, set r← q as a random vector2: β0 ← ∥r∥23: for j ← 1, 2, ..., do4: qj ← r/βj−1
5: r← Aqj
6: r← r− qj−1βj−1
7: αj ← q⊺j r
8: r← r− qjαj
9: r← (I−QjQ⊺j )r,
10: βj ← ∥r∥211: Solve the tridiag problem (Q⊺
j AQj)Ξj = ΘjΞj
12: Form the j approximate eigenvectors QjΞj of A13: If the f top eigenvectors have converged, stop.14: end for15: Compute latent factors V = QfΞ
16: return Π← RVV⊺
Computational Aspects:
• The MV product in j Lanczossteps is O(j · nnz)
• Making the j-th vectororthogonal to the previous onescosts O(jm)
Parallel Implementation:
MovieLens20M
f=50 100 150 200 300
8 cores 11.0 18.0 24.2 28.7 36.816 cores 6.9 11.6 15.6 18.9 24.332 cores 4.8 8.1 11.1 13.5 17.664 cores 3.5 6.0 8.2 9.9 12.5
EigenRec
The Code is available here:https://github.com/nikolakopoulos/EigenRec
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Qualitative Evaluation
Effects of Prior Popularity
Methodology• Randomly sample 1.4% of the ratings of
the dataset ⇒ probe set P• Use each item vj , rated with 5 stars by
user ui in P ⇒ test set T• Randomly select another 1000 unrated
items of the same user for each item in T• Form ranked lists by ordering all the 1001
items
Metrics• Recall• Precision• R-Score• NDCG@k• MRR
−2 −1 0 1 2
0.1
0.2
0.3
Yaho
o
Cosine
PureSVD
−2 −1 0 1 2
0.1
0.2
0.3
Pearson
−2 −1 0 1 2
0.1
0.2
0.3
Jaccard
−2 −1 0 1 20
0.10.20.3
d
ML1M
PureSVD
−2 −1 0 1 20
0.10.20.3
d
−2 −1 0 1 20
0.10.20.3
d
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Standard Top-N Recommendations
Methodology• Randomly sample 1.4% of the ratings of
the dataset ⇒ probe set P• Use each item vj , rated with 5 stars by
user ui in P ⇒ test set T• Randomly select another 1000 unrated
items of the same user for each item in T• Form ranked lists by ordering all the 1001
items
Metrics• Recall• Precision• R-Score• NDCG@k• MRR
5 10 15 200
0.2
0.4
0.6
Yaho
o
Recall
0.2 0.4 0.6 0.8 10
0.1
0.2
Precision vs Recall
5 10 15 200
0.2
0.4
0.6
NDCG@N
5 10 15 200
0.2
0.4
0.6
RScore(α)
5 10 15 200
0.2
0.4
0.6
ML1M
0.2 0.4 0.6 0.8 10
0.1
0.2
5 10 15 200
0.2
0.4
5 10 15 200
0.2
0.4
0.6
EigenRec RCT MD RED L† MFA7
Long-Tail Recommendations
Methodology• We order the items according to their
popularity (measured in terms of numberof ratings)
• We further partition the test set T intotwo subsets, Thead and Ttail
• We discard the popular items and weevaluate EigenRec and the otheralgorithms on the Ttail test set, using theprocedure explained previously.
Metrics• Recall• Precision• R-Score• NDCG@k• MRR
5 10 15 200
0.2
0.4
0.6
Yaho
o
Recall
0.2 0.4 0.6 0.8 10
0.1
0.2
Precision vs Recall
5 10 15 200
0.2
0.4
NDCG@N
5 10 15 200
0.2
0.4
0.6
RScore(α)
5 10 15 200
0.2
0.4
ML1M
0.2 0.4 0.6 0.8 10
.05
0.1
5 10 15 200
0.1
0.2
0.3
5 10 15 200
0.2
0.4
EigenRec RCT MD RED L† MFA8
Cold-Start Recommendations I
Cold-Start Problem
• Difficulty of making reliable recommendations due to an initiallack of ratings
• In beginning stages, when there is not sufficient number of ratingsfor the collaborative filtering algorithms to uncover similarities ⇒New Community Problem
• Introduction of new users to an existing system where they have notrated many items ⇒ New Users Problem
9
Cold-Start Recommendations II
New Community:• Methodology: Randomly select to include 33%, 66%, and 100% of the overall ratings on
two new artificially sparsified versions of the dataset.
33% 66% 100% 33% 66% 100%
0.1
0.2
0.3
MRR
EigenRec RCT MD RED L† MFA
Recommending All Items Recommending Long Tail Items
New Users:• Methodology: Randomly select 50 users having rated at least 100 items and randomly
delete 95% of each users’ ratings.
5 10 15 200
0.1
0.2
0.3
NewU
sers
Full Recall
0.2 0.4 0.6 0.8 10
.02
.04
.06
.08
Precision vs Recall
5 10 15 200
0.1
0.2
NDCG@N
5 10 15 200
0.1
0.2
0.3
RScore(α)
EigenRec RCT MD RED L† MFA10
Conclusions and Future Work
Conclusions and Future Work
EigenRec• Computationally Efficient framework for Top-N Recommendations• Allows for flexible modeling and control of the effects of prior
popularity• Natural generalization of PureSVD
• (+) Optimize its Top-N recommendation performance• (+) Alleviate its inherent popularity bias• (+) Compute it more efficiently
• Good Top-N Recommendation Performance
Future Directions• Explore more elaborate Similarity and Scaling functions• Explore the Hierarchical structure of the Itemspace
11
References
A. N. Nikolakopoulos, V. Kalantzis, E. Gallopoulos, and J. Garofalakis.EigenRec: Generalizing PureSVD for Effective and Efficient Top-NRecommendations.Knowledge and Information Systems, 2018.A. N. Nikolakopoulos, V. Kalantzis, E. Gallopoulos, and J. Garofalakis.Factored Proximity Models for Top-N Recommendations.In 2017 IEEE International Conference on Big Knowledge (ICBK), pp. 80-87,Hefei, 2017.P. Cremonesi, Y. Koren, and R. Turrin.Performance of recommender algorithms on top-n recommendation tasks.In Proceedings of the fourth ACM conference on Recommender systems, RecSys’10, pages 39–46. ACM, 2010.F. Fouss, K. Francoisse, L. Yen, A. Pirotte, and M. Saerens.An experimental investigation of kernels on graphs for collaborativerecommendation and semisupervised classification.Neural Netw., 31:53–72, July 2012.
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Thank you for your Attention!
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Questions?
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