Post on 16-Apr-2017
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