A Kernel-Based Approach to Exploiting Interaction-Networks in Heterogeneous Information Sources for Improved Recommender Systems Oluwasanmi (Sanmi) Koyejo Joydeep Ghosh ECE Dept., University of Texas at Austin HetRec ’11, October 27, 2011, Chicago, IL, USA Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 1 / 20
Outline Motivation 1 Modeling Approach 2 Experiments 3 Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 2 / 20
Outline Motivation 1 Modeling Approach 2 Experiments 3 Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 3 / 20
Interaction Networks courtesy flickr/yankeeincanada Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 3 / 20
Interaction Networks and Recommender Systems *Proposed approaches include [Golbeck, 2005, Jamali and Ester, 2009, Ma et al., 2008, Jamali and Ester, 2010] Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 4 / 20
Many New Interactions Social network(s) User implicit feedback Item category Item history blog.spoongraphics.co.uk Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 5 / 20
Need a New Approach Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 6 / 20
Need a New Approach Is the data always helpful? Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 6 / 20
Need a New Approach Is the data always helpful? data may be noisy Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 6 / 20
Need a New Approach Is the data always helpful? data may be noisy data may be corrupted or inaccurate Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 6 / 20
Need a New Approach Is the data always helpful? data may be noisy data may be corrupted or inaccurate data may be descriptive, but uncorrelated with target task Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 6 / 20
How To Select the Useful Information? Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 7 / 20
How To Select the Useful Information? Use everything! noisy or uncorrelated data may confuse model Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 7 / 20
How To Select the Useful Information? Use everything! noisy or uncorrelated data may confuse model Use domain knowledge might miss hidden correlations Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 7 / 20
How To Select the Useful Information? Use everything! noisy or uncorrelated data may confuse model Use domain knowledge might miss hidden correlations Goal A model based approach for extracting useful information from multiple interaction networks. Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 7 / 20
Outline Motivation 1 Modeling Approach 2 Experiments 3 Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 8 / 20
Linear Prediction Model z i , j is observed preference of the i th user for the j th item User feature: x i Item feature: y j Linear prediction model: z i , j = w ⊤ ( x i ⊗ y j ) ˆ = x ⊤ i Wy j . Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 8 / 20
Generalized Matrix Factorization Enforce that W have some maximum rank R W = UV ⊤ Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 9 / 20
Generalized Matrix Factorization Enforce that W have some maximum rank R W = UV ⊤ Resulting prediction: z i , j = x ⊤ i UV ⊤ y j ˆ Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 9 / 20
Generalized Matrix Factorization Enforce that W have some maximum rank R W = UV ⊤ Resulting prediction: z i , j = x ⊤ i UV ⊤ y j ˆ Aside: if features are user and item indices e i , e j : z i , j = e ⊤ i UV ⊤ e j = u i ⊤ v j ˆ Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 9 / 20
Extract Interaction Graph Features Graph G = ( V , E , A ) V = { v i } represents the set of entities as vertices E = { e i , j } represents the set of links A represents strength of links. Assume a i , j > 0, a i , j = a j , i . Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 10 / 20
Extract Interaction Graph Features Graph G = ( V , E , A ) V = { v i } represents the set of entities as vertices E = { e i , j } represents the set of links A represents strength of links. Assume a i , j > 0, a i , j = a j , i . Smooth Functions on the graph f : V �→ R f is smooth on G if the average difference ( f ( v i ) − f ( v j )) 2 is small for close points v i and v j [Belkin et al., 2006]. Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 10 / 20
Normalized Laplacian L = D − 1 2 ( D − A ) D − 1 2 = I − D − 1 2 AD − 1 2 D is diagonal di , i = � j ai , j Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 11 / 20
Normalized Laplacian L = D − 1 2 ( D − A ) D − 1 2 = I − D − 1 2 AD − 1 2 D is diagonal di , i = � j ai , j Eigenvectors of L are smooth on G [Smola and Kondor, 2003] Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 11 / 20
Normalized Laplacian L = D − 1 2 ( D − A ) D − 1 2 = I − D − 1 2 AD − 1 2 D is diagonal di , i = � j ai , j Eigenvectors of L are smooth on G [Smola and Kondor, 2003] Eigen-decompose N � ⊤ λ j η j η j L = j = 1 Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 11 / 20
Augmented Features Compute first D k eigenvectors of L from each G k Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 12 / 20
Augmented Features Compute first D k eigenvectors of L from each G k Append features: � Identity e i η 1 ( i ) � Eigenvector G 1 x i = . . . . . . η K ( i ) � Eigenvector G K Same for item side to compute y j Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 12 / 20
Augmented Features Compute first D k eigenvectors of L from each G k Append features: � Identity e i η 1 ( i ) � Eigenvector G 1 x i = . . . . . . η K ( i ) � Eigenvector G K Same for item side to compute y j Expensive! dimension grows with number of basis vectors D k and number of graphs K . Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 12 / 20
Linear Kernels Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 13 / 20
Linear Kernels K p ( i , j ) = x i , p x j , p and G q ( i , j ) = y i , q y j , q Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 13 / 20
Linear Kernels K p ( i , j ) = x i , p x j , p and G q ( i , j ) = y i , q y j , q Separate kernel for each dimension x p and y q Let K 0 = I and G 0 = I Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 13 / 20
Linear Kernels K p ( i , j ) = x i , p x j , p and G q ( i , j ) = y i , q y j , q Separate kernel for each dimension x p and y q Let K 0 = I and G 0 = I Kernel uses R × Nx user factor parameters vs. R × � k Dk for linear model Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 13 / 20
Linear Kernels K p ( i , j ) = x i , p x j , p and G q ( i , j ) = y i , q y j , q Separate kernel for each dimension x p and y q Let K 0 = I and G 0 = I Kernel uses R × Nx user factor parameters vs. R × � k Dk for linear model Combine: P Q � � K = a p K p and G = a q G q p = 1 q = 1 � � { a p ≥ 0, a p = 1} , { b q ≥ 0, b q = 1} p q Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 13 / 20
Outline Motivation 1 Modeling Approach 2 Experiments 3 Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 14 / 20
Lastfm Data 0.35 0.30 0.25 User-Artist Listen counts Probability 0.20 N x = 1982, N y = 17632, N = 92834 0.15 Interaction: P = 151, Q = 101: 0.10 User Social network D = 50 0.05 User-Tag-Artist graph; 0.00 converted to user-user D = 100 0 2 4 6 8 10 12 14 Log(Listen count) and item-item D = 100 Figure: Histogram of log transformed listen counts in Last.fm µ = 5.469, σ = 1.531. Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 14 / 20
Lastfm Results Rank=5 Rank=10 Global Model 1.502 (0.014) - Uniform Interaction 1.492 (0.015) 1.498 (0.008) MF 1.139 (0.006) 1.173 (0.010) Weighted Interaction 1.071 (0.009) 1.106 (0.006) Table: Average (std.) cross validation RMSE on Last.fm 1 1 Uniform: a p = P + 1 , b q = Q + 1 MF: a 0 = 1, b 0 = 1 Koyejo & Ghosh (UT Austin) Kernels for Interaction Networks HetRec ’11 15 / 20
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