Presenter: Jose Lugo Pedja’s Lab Meetingmontana.informatics.indiana.edu/LabWebPage/... ·...
Transcript of Presenter: Jose Lugo Pedja’s Lab Meetingmontana.informatics.indiana.edu/LabWebPage/... ·...
Weisfeiler-Lehman Graph Kernel (JMLR 2011)and
Neighborhood Hash Graph Kernel (ICDM 2009)
Presenter: Jose LugoPedja’s Lab Meeting
October 12, 2011
Learning on Graphs• Application domains
– Bioinformatics, Cheminformatics, WWW link, Social networks
• Motivation: Study relationships between structured objects (graphs)
G G’Graph Comparison Problem
• Define kernels on pair of graphs
k(G, G’) = <Φ(G), Φ(G’)>
k(G, G’) – measure of similarity between G and G’
• Kernel Matrix K, where Kij = k(Gi, Gj) for 1 ≤ i,j ≤ n– Properties of K
I. Symmetric II. Positive semi-definite
Graph Kernels
k(G1, G2) = <Φ(G1), Φ(G2)> = 15
G1 G2
Φ(G) = (#(T), #(L))T :=
L :=
Φ(G1) = (1, 3) Φ(G2) = (0, 5)
Graph Kernel Example
Graph Kernels
Random WalkKernels
AlgebraicKernels
Rational Kernels1
1. Certain Rational Kernels when specialized to graphs reduced to Random Walk Graph Kernel (Vishwanathan et. al. 2010)
Gӓrtner et. al. (2003)Borgwardt et. al. (2005)
Vishwanathan et. al. (2006)
Cortes et. al. (2002,2003, 2004)
Tsuda et. al. (2002)Kashima et. al. (2003,2004)
Mahé et. al. (2004)
Kondor & Borgwardt (2008)
Graphlet Kernels
Borgwardt et. al. (2007)Shervashidze et. al. (2009)
Vacic et. al. (2010)
Graph Kernels Research Efforts
MarginalizedKernels
Other GraphKernels
Ramon and Gӓrtner (2003)Horváth et. al. (2004)
Ralaivola et. al. (2005)Frӧhlich et. al. (2005)
Menchetti et. al. (2005)Borgwardt et. al. (2005)Mahé and Vert (2008)
Reviewed onVishwanathan et. al. (2010)
Kondor & Lafferty (2002)
Image taken from “A Linear-time Graph Kernel” talk by Shohei Hido, IEEE ICDM2009, Miami, Florida, 12/09/2009
Question: How to scale up graph kernels to large, labeled graphs?
f: V → Σ = {A, R, N, D, C, E, Q, G, H, I, L, K, M, F, P, S, T, W, T, Y, V}
Graph Kernels on (Large) Labeled Graphs
f: V → Σ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20}
or
G = (V, E, f), |V| = n, |E| = m
Weisfeiler-Lehman Graphlet KernelShervashidze et. al. (2010)
• Weisfeiler-Lehman test of isomorphism (1968)
• Define Weisfeiler-Lehman graph kernels: – kWL(G, G’), kWLsubtree(G, G’) and kWLshortestpath(G, G’)
Observed that compressed labels li(v) correspond to subtree patterns of height i rooted at v
Example
Φ(G0) = (2, 1, 1, 1, 1)Φ(G0’) = (1, 2, 1, 1, 1)k(G0, G0’) = < Φ(G0) , Φ(G0’)> = 7
Φ(G1) = (2, 0, 1, 0, 1, 1, 0, 1)Φ(G1’) = (1, 1, 0, 1, 1, 0, 1, 1)k(G1, G1’) = < Φ(G1) , Φ(G1’)> = 4
Example
Neighborhood Hash Graphlet KernelHido and Kashima (2009)
• Bit-represented node label
• Logical operations
• Neighborhood hash over nodes
• Define neighborhood hash graph kernel, kNH(G, G’)
• Linear time complexity with # of edges
Bit-represented Node Label
Image taken from “A Linear-time Graph Kernel” talk by Shohei Hido, IEEE ICDM2009, Miami, Florida, 12/09/2009
Logical Operations on Bit Labels
• XOR (si, sj)– Exclusive OR– Order-independent
• ROTk– k-bit rotation– move left most k-bits to
the right
Neighborhood Hash over a Node, NH(v)
NH(v) uniquely represents the distribution of the node labels around v
Neighborhood Hash over a Graph, NH(G)
G0
Gi = NH(Gi-1)GiG1 …
ith-Hash graph1st-Hash graph
Image taken from “A Linear-time Graph Kernel” talk by Shohei Hido, IEEE ICDM2009, Miami, Florida, 12/09/2009
Gr+1 contains high-order relationships between the nodes with order r
Neighborhood Hash Graph Kernel K(i)
NH(Gi, Gi’)
Example
Image taken from “A Linear-time Graph Kernel” talk by Shohei Hido, IEEE ICDM2009, Miami, Florida, 12/09/2009
QUESTIONS?
Thank You!