2018/07/30 Deep Learning JP: http://deeplearning.jp/hacks/

1. Graph Convolutional Network
LT
M2

2. • Graph NN
• Graph
•
•
•
• GCN

3. • Graph Convolution
• Graph Fourier
• Graph Convolution
• Graph Convolutional Network
• 2
•
•
•

4. [1]

5. Graph Convolution
• Convolution Graph
•
• 2
• Graph Fourier
•
•
•
• Graph Fourier
•

6. Graph Convolution ( )
•
…
↓
Convolution Theorem ( Fourier )
↓
Graph Graph Fourier
Convolution Theorem
Convolution Theorem
!" ∗ $ = &" ⊙ ($
&" : f Fourier
* :
⊙:

7. Graph Fourier
•
•
• G = (V, E, W)
• V: , E: , W: (i, j) (i,j)
• d (d = 1)
• Graph Fourier
•

8. Graph Fourier
•
• ! ∈ ℝ$
• ! 1 %&
'
(,*
+(,* ,( − ,*
.
• %$/0
• %&, %0, … %$/0 ℝ$
%(20 = argmin
: ∈ ℝ;, : <0, :=>?…=>@
'
(,*
+(,* ,( − ,*
.

9. •
!
",$
%",$ &" − &$
(
= 2&+
,&
• -., -/, … -12/ L

10. Graph Fourier
• ! → #! = % = &', &), … &+,) - GF .-
! = /
0
%010
• U = 1', 1), … 1+,) Graph Fourier
#! = 23
!
• Graph Fourier
4#! = 2#!

11. Graph Convolution
• Graph Convolution / ∗1 2
1. /, 2 GF : / → 8/ = :;
/
2. 8= : 8/ ⊙ 8=
3. GF : ?8/ ⊙ 8= = :(8/ ⊙ 8=)
• Convolution Theorem
/ ∗1 2 = :(8/ ⊙ 8=) = U(:;
/ ⊙ :;
=)
• GCN BC= DEFB(G)
BC ∗1 I: = :BC:;
I
Convolution Theorem
?J ∗ B = KJ ⊙ LB
KJ : f Fourier
* :
⊙:

12. Graph Convolution
• ! = ($%, $', … $)*')
1. . GF : . → 1. = 23
.
2. ! : ! ⊙ 1.
3. GF : 6! ⊙ 1. = 2(! ⊙ 1.)
• Convolution Theorem
! ∗ . ∶= 2(! ⊙ 1.) = U(! ⊙ 23
.)
• ! ⊙ 23
. = 9:;< ! 23
.
<= ∗ >: = 2<=23
>
Convolution Theorem
6? ∗ < = @? ⊙ A<
@? : f Fourier
* :
⊙:

13. • Graph Convolution (!" = $%&!(())
!" ∗+ ,: = .!"./
,
•
•
• CNN
•
• U : O(n^2)
•

14. 2 Graph Convolutional Network
• [2]
• [2] [3]

15. • !" L Λ
[2]
• K K
• O(|E|)

16. • K=1
[3]
! = #$%&
! ∈ ℝ) × +
:
% ∈ ℝ) × ,
: ( )
& ∈ ℝ, × +
:
#$ ∈ ℝ, × +
:
GCN

17. • GCN

18. • Karate club
• 4
•
[6]

19. WordNet
• Wang, et al [4]
• GCN(Graph Convolutional Network) Zero-shot Learning
• Chen, et al [5]
• GCN

20. • Graph Convolution
• Graph Fourier Convolution Theorem
• Graph Convolution Network
•
•
• GCN

21. [1]: https://tech-blog.abeja.asia/entry/2017/04/27/105613
• GCN
[2] Michael et al. Convolutional neural networks on graphs with fast localized spectral
filtering. NIPS’2016.
[3] Thomas et al. Semi-Supervised Classification with Graph Convolutional Networks.
ICLR’17
[4] X. Wang, et al. Zero-shot Recognition via Semantic Embeddings and Knowledge Graphs.
CVPR’18
[5] Meihao Chen, et al. Graph convolutional networks for classification with a structured
label space. arXiv preprint arXiv:1710.04908, 2018
[6] https://tkipf.github.io/graph-convolutional-networks/
• [3]