NCDREC: A Decomposability Inspired Framework for Top-N Recommendation

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1 NCDREC: A Decomposability Inspired Framework for Top-N Recommendation Athanasios N. Nikolakopoulos,2 John D. Garofalakis,2 Computer Engineering and Informatics Department, University of Patras, Greece 2 Computer Technology Institute & Press Diophantus IEEE/WIC/ACM International Conference on Web Intelligence Warsaw, August 24

2 Outline Introduction & Motivation Introduction & Motivation Recommender Systems - Collaborative Filtering Challenges of Modern CF Algorithms 2 NCDREC Model Criterion for ItemSpace Coverage NCDREC Algorithm: Storage and Computational Issues 3 Evaluation Methodology Quality of Recommendations Long-Tail Recommendations Cold-Start Recommendations 4

3 Introduction & Motivation Recommender Systems - Collaborative Filtering Challenges of Modern CF Algorithms Recommender System Algorithms Recommendation USERS ITEMS List RECOMMENDER. SYSTEM Rating Predictions RATINGS... Collaborative Filtering Recommendation Algorithms Wide deployment in Commercial Environments Significant Research Efforts A. N. Nikolakopoulos and J. D. Garofalakis NCDREC

4 Challenges of Modern CF Algorithms Recommender Systems - Collaborative Filtering Challenges of Modern CF Algorithms Sparsity It is an Intrinsic RS Characteristic related to serious problems: Long-Tail Recommendation Cold start Problem Limited ItemSpace Coverage Traditional CF techniques, such as neighborhood models, are very susceptible to sparsity. Among the most promising approaches in alleviating sparsity related problems are: Dimensionality-Reduction Models Build a reduced latent space which is dense. Graph-Based Models. Exploit transitive relations in the data, while preserving some of the locality.

5 Exploiting Decomposability Recommender Systems - Collaborative Filtering Challenges of Modern CF Algorithms We attack the problem from a different perspective: Sparsity Hierarchy Decomposability. Nearly Completely Decomposable Systems Pioneered by Herbert A. Simon. Many Applications in Diverse Disciplines such as economics, cognitive theory and social sciences, to computer systems performance evaluation, data mining and information retrieval Main Idea: Exploit the innate Hierarchy of the Item Set, and view it as a decomposable space. Can this enrich the Collaborative Filtering Paradigm in an Efficient and Scalable Way? Does this approach offer any qualitative advantages in alleviating sparsity related problems?

6 NCDREC Model Overview NCDREC Model Criterion for ItemSpace Coverage NCDREC Algorithm: Storage and Computational Issues Definitions We define a D-decomposition to be an indexed family of sets D {D,..., D K }, that span the ItemSpace V, We define D v v D k D k to be the proximal set of items of v V, We also define the associated block coupling graph G D (V D, E D ); its vertices correspond to the D-blocks, and an edge between two vertices exists whenever the intersection of these blocks is a non-empty set. Finally, we introduce an aggregation matrix A D R m K, whose jk th element is, if v j D k and zero otherwise. NCDREC Components Main Component: Recommendation vectors produced by projecting the NCD perturbed data onto an f -dimensional space. ColdStart Component:Recommendation vectors are the stationary distributions of a Discrete Markov Chain Model. G R + ɛw W ɛzx X diag(a D e) A D [Z] ik (n k u ) [RA D ] ik, when n k i u >, i and zero otherwise. S(ω) ( α)e + α(βh + ( β)d) H diag(ce) C, where [C] ij r i r j for i j D XY, X diag(a D e) A D, Y diag(a D e) A D, E eω

7 Criterion for ItemSpace Coverage NCDREC Model Criterion for ItemSpace Coverage NCDREC Algorithm: Storage and Computational Issues Theorem (ItemSpace Coverage) If the block coupling graph G D is connected, there exists a unique steady state distribution π of the Markov chain corresponding to matrix S that depends on the preference vector ω; however, irrespectively of any particular such vector, the support of this distribution includes every item of the underlying space. Proof Sketch: When G D is connected, the Markov chain induced by the stochastic matrix S consists of a single irreducible and aperiodic closed set of states, that includes all the items. The above is true for every stochastic vector ω, and for every positive real numbers α, β <. Taking into account the fact that the state space is finite, the resulting Markov chain becomes ergodic. So π i >, for all i, and the support of the distribution that defines the recommendation vector includes every item of the underlying space.

8 NCDREC Model Criterion for ItemSpace Coverage NCDREC Algorithm: Storage and Computational Issues NCDREC Algorithm: Storage and Computational Issues Input: Matrices R R n m, H R m m, X R m K, Y R K m, Z R n K. Parameters α, β, f, ɛ Output: The matrix with recommendation vectors for every user, Π R n m Step : Find the newly added users and collect their preference vectors into matrix Ω. Step 2: Compute Π sparse using the ColdStart Procedure. Step 3: Initialize vector p to be a random unit length vector. Step 4: Compute the modified Lanczos procedure up to step M, using NCD PartialLBD with starting vector p. Step 5: Compute the SVD of the bidiagonal matrix B and use it to extract f < M approximate singular triplets: {ũ j, σ j, ṽ j } {Qu (B) j, σ (B) j, Pv (B) j } Step 6: Orthogonalize against the approximate singular vectors to get a new starting vector p. Step 7: Continue the Lanczos procedure for M more steps using the new starting vector. Step 8: Check for convergence tolerance. If met compute matrix: Π full = ŨΣṼ else go to Step 4 Step 9: Update Π full, replacing the rows that correspond to new users with Π sparse. Return Π full

9 Evaluation Methodology Quality of Recommendations Long-Tail Recommendations Cold-Start Recommendations Datasets Yahoo!R2Music MovieLens Competing Methods Commute Time (CT) Pseudo-Inverse of the user-item graph Laplacian (L ) Matrix Forest Algorithm (MFA) First Passage Time (FP) Katz Algorithm (Katz) Metrics Recall Precision R-Score Normalized Discounted Cumulative Gain (NDCG@k) Mean Reciprocal Rank

10 Full Dist. Recommendations Methodology Randomly sample.4% of the ratings of the dataset probe set P Use each item v j, rated with 5 stars by user u i in P test set T Randomly select another unrated items of the same user for each item in T Form ranked lists by ordering all the items Evaluation Methodology Quality of Recommendations Long-Tail Recommendations Cold-Start Recommendations TABLE I RECOMMENDATION QUALITY ON MOVIELENSM AND YAHOO!R2MUSIC DATASETS USING R-SCORE AND MRR METRICS MovieLensM Yahoo!R2Music R(5) R() MRR R(5) R() MRR NCDREC MFA L FP Katz CT Recall@N Precision/Recall NDCG@N MovieLensM Yahoo!R2Music NCDREC Katz FP MFA L CT

11 Long-Tail Recommendations Methodology (Long Tail) We order the items according to their popularity (measured in terms of number of ratings) We further partition the test set T into two subsets, T head and T tail We discard the popular items and we evaluate NCDREC and the other algorithms on the T tail test set. Evaluation Methodology Quality of Recommendations Long-Tail Recommendations Cold-Start Recommendations TABLE II LONG TAIL RECOMMENDATION QUALITY ON MOVIELENSM AND YAHOO!R2MUSIC DATASETS USING R-SCORE AND MRR METRICS MovieLensM Yahoo!R2Music R(5) R() MRR R(5) R() MRR NCDREC MFA L FP Katz CT Recall@N Precision/Recall NDCG@N MovieLensM Yahoo!R2Music NCDREC Katz FP MFA L CT

12 New Users problem Introduction & Motivation Evaluation Methodology Quality of Recommendations Long-Tail Recommendations Cold-Start Recommendations Methodology Randomly select users having rated at least items and delete 96%, 94%, 92% and 9% of each users ratings. Compare the rankings induced on the modified data with the complete set of ratings. Metrics Spearman s ρ Kendall s τ Degree of Agreement (DOA) Normalized Distance-based Performance Measure (NDPM) Spearman s ρ Kendall s τ % 6% 8% % Percentage of included ratings Degree Of Agreement % 6% 8% % Percentage of included ratings NDPM (the smaller the better) % 6% 8% % 4% 6% 8% % Percentage of included ratings Percentage of included ratings NCDREC Katz FP MFA L CT

13 Future Research Directions & Conclusion Future Work Decomposition Granularity Effect Coarse Grained Sparseness Insensitivity Fine Grained Higher Quality of Recommendations Multiple-Criteria Decompositions How it affects the theoretical properties of the ColdStart Component?

14 Thanks! Q&A

15 /3 /2 /2 /4 /4 /4 /4 /2 /5 /5 /2 /2 Example NCD Proximity Matrix NCDREC Basic SubComponents Example NCD Proximity Matrix Back D D 2 D 3 N v G v v {v, v 2, v 4} v 2 2 {v, v 2, v 4, v 5, v 6, v 7, v 8} v 3 {v 3, v 4, v 5, v 8} v 4 2 {v, v 2, v 3, v 4, v 5, v 8} v 5 2 {v 2, v 3, v 4, v 5, v 6, v 7, v 8} v 6 {v 2, v 5, v 6, v 7, v 8} v 7 {v 2, v 5, v 6, v 7, v 8} v 8 2 {v 2, v 3, v 4, v 5, v 6, v 7, v 8} /2 /2 /2 v v2 v3 v4 v5 v6 v7 v8 D D2 D3 /3 /3 /5 /5 /5 v v2 v3 v4 v5 v6 v7 v8 = D =

16 Example NCD Proximity Matrix NCDREC Basic SubComponents NCD PartialLBD Procedure ColdStart Procedure NCD PartialLBD Back procedure NCD PartialLBD(R, X, Z, p, ɛ) φ X p ; q Rp + ɛzφ; b, q 2 ; u q /b, ; for j = to M do φ Z q j ; r R q j + ɛxφ b j,j p j ; r r [p... p j ] ([p... p j ] r); if j < M then b j,j+ r ; p j+ r/b j,j+ ; φ X p j+ ; q j+ Rp j+ + ɛzφ b j,j+ q j ; q j+ q j+ [q... q j ] ([q... q j ] q j+ ); b j+,j+ q j+ ; q j+ q j+ /b j+,j+ ; end if end for end procedure

17 Example NCD Proximity Matrix NCDREC Basic SubComponents NCD PartialLBD Procedure ColdStart Procedure ColdStart Procedure Back procedure ColdStart(H, X, Y, Ω, α, β) Π Ω; k ; r ; while r > tol and k maxit do k k + ; ˆΠ αβπh; Φ ΠX; ˆΠ ˆΠ + α( β)φy + ( α)ω; r ˆΠ Π ; Π ˆΠ; end while return Π sparse Π end procedure

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