Bipartite Edge Prediction via Transductive Learning over Product - PowerPoint PPT Presentation

Bipartite Edge Prediction via Transductive Learning over Product Graphs Bipartite Edge Prediction via Transductive Learning over Product Graphs Hanxiao Liu, Yiming Yang School of Computer Science, Carnegie Mellon University July 8, 2015 ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 1

Bipartite Edge Prediction via Transductive Learning over Product Graphs Problem Description Outline 1 Problem Description 2 The Proposed Framework 3 Formulation Product Graph Construction Graph-based Transductive Learning 4 Optimization 5 Experiment 6 Conclusion ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 2

Bipartite Edge Prediction via Transductive Learning over Product Graphs Problem Description Problem Description Many applications involve predicting the edges of a bipartite graph. A 1 Recommender System -2 I ? 2 Host-Pathogen Interaction ? B 3 Question-Answering Mapping +5 ? II 4 Citation Network . . . ? C

Bipartite Edge Prediction via Transductive Learning over Product Graphs Problem Description Problem Description Many applications involve predicting the edges of a bipartite graph. A 1 Recommender System -2 I ? 2 Host-Pathogen Interaction ? Graph G B Graph H 3 Question-Answering Mapping +5 ? II 4 Citation Network . . . ? C Sometimes, vertex sets on both sides are intrinsically structured. ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 4

Bipartite Edge Prediction via Transductive Learning over Product Graphs Problem Description Problem Description Many applications involve predicting the edges of a bipartite graph. A 1 Recommender System -2 I ? 2 Host-Pathogen Interaction ? Graph G B Graph H 3 Question-Answering Mapping +5 ? II 4 Citation Network . . . ? C Sometimes, vertex sets on both sides are intrinsically structured. Heterogeneous info: G + H + partial observations ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 5

Bipartite Edge Prediction via Transductive Learning over Product Graphs Problem Description Problem Description Many applications involve predicting the edges of a bipartite graph. A 1 Recommender System -2 I ? 2 Host-Pathogen Interaction ? Graph G B Graph H 3 Question-Answering Mapping +5 ? II 4 Citation Network . . . ? C Sometimes, vertex sets on both sides are intrinsically structured. Heterogeneous info: G + H + partial observations Combine them to make better edge predictions? ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 6

Bipartite Edge Prediction via Transductive Learning over Product Graphs The Proposed Framework The Proposed Framework A -2 I ? ? B Graph G Graph H +5 ? II ? C Transductive learning should be effective 1 Labeled edges (red) are highly sparse 2 Unlabeled edges (gray) are massively available ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 7

Bipartite Edge Prediction via Transductive Learning over Product Graphs The Proposed Framework The Proposed Framework A -2 I ? ? B Graph G Graph H +5 ? II ? C Transductive learning should be effective 1 Labeled edges (red) are highly sparse 2 Unlabeled edges (gray) are massively available Assumption: similar edges should have similar labels ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 8

Bipartite Edge Prediction via Transductive Learning over Product Graphs The Proposed Framework The Proposed Framework A -2 I ? ? B Graph G Graph H +5 ? II ? C Transductive learning should be effective 1 Labeled edges (red) are highly sparse 2 Unlabeled edges (gray) are massively available Assumption: similar edges should have similar labels Prerequisite: a similarity measure among the edges, i.e. a “Graph of Edges” (not directly provided) ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 9

Bipartite Edge Prediction via Transductive Learning over Product Graphs The Proposed Framework The Proposed Framework A -2 I ? ? B Graph G Graph H +5 ? II ? C Transductive learning should be effective 1 Labeled edges (red) are highly sparse 2 Unlabeled edges (gray) are massively available Assumption: similar edges should have similar labels Prerequisite: a similarity measure among the edges, i.e. a “Graph of Edges” (not directly provided) Can be induced from G and H via Graph Product! ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 10

Bipartite Edge Prediction via Transductive Learning over Product Graphs The Proposed Framework The Proposed Framework The “Graph of Edges” can be induced by taking the product of G and H In the product graph G ◦ H Each Vertex ∼ edge (in the original bipartite graph) Each Edge ∼ edge-edge similarity ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 11

Bipartite Edge Prediction via Transductive Learning over Product Graphs The Proposed Framework The Proposed Framework The “Graph of Edges” can be induced by taking the product of G and H In the product graph G ◦ H Each Vertex ∼ edge (in the original bipartite graph) Each Edge ∼ edge-edge similarity The adjacency matrix of the product graph is defined by “ ◦ ” (to be discussed later). ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 12

Bipartite Edge Prediction via Transductive Learning over Product Graphs The Proposed Framework The Proposed Framework Problem Mapping Edge Prediction Vertex Prediction (Original Problem) (Equivalent Problem) Given G , H and labeled edges, Given G ◦ H and labeled vertices, predict the unlabeled edges predict the unlabeled vertices ? -2 ? A (I , C ) (I , A ) (I , B ) -2 I ? ? B ? ? +5 +5 ? II (II , C ) (II , A ) (II , B ) ? C ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 13

Bipartite Edge Prediction via Transductive Learning over Product Graphs Formulation Outline 1 Problem Description 2 The Proposed Framework 3 Formulation Product Graph Construction Graph-based Transductive Learning 4 Optimization 5 Experiment 6 Conclusion ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 14

Bipartite Edge Prediction via Transductive Learning over Product Graphs Formulation Product Graph Construction Outline 1 Problem Description 2 The Proposed Framework 3 Formulation Product Graph Construction Graph-based Transductive Learning 4 Optimization 5 Experiment 6 Conclusion ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 15

Bipartite Edge Prediction via Transductive Learning over Product Graphs Formulation Product Graph Construction Product Graph Construction Q: When should vertex ( i, j ) ∼ ( i ′ , j ′ ) in the product graph? Tensor GP i ∼ i ′ in G AND j ∼ j ′ in H � i ∼ i ′ in G AND j = j ′ � � i = i ′ AND j ∼ j ′ in H � Cartesian GP OR ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 16

Bipartite Edge Prediction via Transductive Learning over Product Graphs Formulation Product Graph Construction Product Graph Construction Q: When should vertex ( i, j ) ∼ ( i ′ , j ′ ) in the product graph? Tensor GP i ∼ i ′ in G AND j ∼ j ′ in H � i ∼ i ′ in G AND j = j ′ � � i = i ′ AND j ∼ j ′ in H � Cartesian GP OR Can be trivially generalized to weighted graphs. ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 17

Bipartite Edge Prediction via Transductive Learning over Product Graphs Formulation Product Graph Construction Product Graph Construction Q: When should vertex ( i, j ) ∼ ( i ′ , j ′ ) in the product graph? Tensor GP i ∼ i ′ in G AND j ∼ j ′ in H � i ∼ i ′ in G AND j = j ′ � � i = i ′ AND j ∼ j ′ in H � Cartesian GP OR Can be trivially generalized to weighted graphs. To compute the adjacency matrices of PG G ◦ T ensor H = G ⊗ H � �� � Kronecker (a.k.a. Tensor) Product G ◦ Cartesian H = G ⊗ I + I ⊗ H = G ⊕ H � �� � Kronecker Sum ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 18

Bipartite Edge Prediction via Transductive Learning over Product Graphs Formulation Product Graph Construction Product Graph Construction Both GPs can be written in the form of spectral decomposition � ( λ i × µ j )( u i ⊗ v j )( u i ⊗ v j ) ⊤ G ◦ T ensor H = (1) i,j � ( λ i + µ j )( u i ⊗ v j )( u i ⊗ v j ) ⊤ G ◦ Cartesian H = (2) i,j ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 19

Bipartite Edge Prediction via Transductive Learning over Product Graphs Formulation Product Graph Construction Product Graph Construction Both GPs can be written in the form of spectral decomposition � ( u i ⊗ v j )( u i ⊗ v j ) ⊤ G ◦ T ensor H = ( λ i × µ j ) (1) � �� � i,j soft AND � ( u i ⊗ v j )( u i ⊗ v j ) ⊤ G ◦ Cartesian H = ( λ i + µ j ) (2) � �� � i,j soft OR The interplay of graphs is captured by the interplay of their spectrum! ICML 2015 Bipartite Edge Prediction via Transductive Learning over Product Graphs 20