The Multiscale Laplacian Graph Kernel

Risi Kondor Department of Computer Science and

Department of Statistics, University of Chicago

Horace Pan Department of Computer Science,

University of Chicago

B4

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NIPS 2016

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�(x)

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�(x)

�(x)

k(xi, xj) = ��(xi), �(xj)�

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k(xi, xj)

�(x)?

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�(x)

k(xi, xj)

�(x)

k

k(xi, xj) = ��(xi), �(xj)�

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k(xi, xj)

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The Multiscale Laplacian Graph Kernel Risi Kondor : University of Chicago Horace Pan : University of Chicago

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global structure

local structure

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Multiscale Laplacian Graph Kernel MLG

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graph Laplacian LG

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LG

LG

wi,j

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LG

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LG

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LG

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LG

vj �(vj)

U = [�(v1), �(v2), . . . , �(vn)]

UL�1UTUL�1UT

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LGLG

LG

kLG

kFLG

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LGLG

LG

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LG

LG l Gl

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LG

LG

} LG

l Gl

kFLG(Gl, G�l)

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LG

LG

} kFLG(Gl, G�l)

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LG

LG

} Kl(v, v�)

kFLG(Gl(v), Gl(v�))

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LG

→

} Kl(v, v�)

Kl(v, v�) l

kFLG(Gl(v), Gl(v�))

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LG

}Kl(v, v�)

l + 1

kFLG(Gl+1(v), Gl+1(v�))

l

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LG

}Kl(v, v�)

l + 1 Kl+1(v, v�)

Kl+1(v, v�)

kFLG(Gl+1(v), Gl+1(v�))

l

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LG

}Kl(v, v�)

l + 1 Kl+1(v, v�)

Kl+1(v, v�)

l

kKlFLG(Gl+1(v), Gl+1(v

�))

Kll

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LG

ll = 0, 1, 2, . . . , L

l = 0, 1, 2, . . . , Ll = 0, 1, 2, . . . , Ll = 0, 1, 2, . . . , L l = 0, 1, 2, . . . , L

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LG

ll = 0, 1, 2, . . . , L

l = 0, 1, 2, . . . , Ll = 0, 1, 2, . . . , Ll = 0, 1, 2, . . . , L l = 0, 1, 2, . . . , L

Kl(v, v�) = kKl�1

FLG(Gl(v), Gl(v�))

l

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LG

Multiscale Laplacian Graph Kernel

ll = 0, 1, 2, . . . , L

l = 0, 1, 2, . . . , Ll = 0, 1, 2, . . . , Ll = 0, 1, 2, . . . , L l = 0, 1, 2, . . . , L

K(G1, G2) = kKLFLG(G1, G2)

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ENZYMES dataset

600 32 16 2

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SVM

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SVM

some of top performance graph kernels Weisfeiler-Lehman Kernel Weisfeiler-Lehman Edge Kernel Shortest Path Kernel Graphlet Kernel p-random Walk Kernel

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NCI1, NCI109

Weisfeiler Lehman / Weisfeiler Lehman Edge Kernel

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LG FLG kernel

LG MLG kernel

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multiresolution structure

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Appendix

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�(x)

��(xi), �(xj)� = k(xi, xj)

k(xi, xj)

�(x)

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LG