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[DL Hacks]Graph Convolutional Network LT

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Deep Learning JP
Deep Learning JP

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

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Graph Convolutional Network LT M2
• Graph NN • Graph • • • • GCN
• Graph Convolution • Graph Fourier • Graph Convolution • Graph Convolutional Network • 2 • • •
[1]
Graph Convolution • Convolution Graph • • 2 • Graph Fourier • • • • Graph Fourier •
Graph Convolution ( ) • … ↓ Convolution Theorem ( Fourier ) ↓ Graph Graph Fourier Convolution Theorem Convolution Theorem !" ∗ $ = &" ⊙ ($ &" : f Fourier * : ⊙:
Graph Fourier • • • G = (V, E, W) • V: , E: , W: (i, j) (i,j) • d (d = 1) • Graph Fourier •
Graph Fourier • • ! ∈ ℝ$ • ! 1 %& ' (,* +(,* ,( − ,* . • %$/0 • %&, %0, … %$/0 ℝ$ %(20 = argmin : ∈ ℝ;, : <0, :=>?…=>@ ' (,* +(,* ,( − ,* .
• ! ",$ %",$ &" − &$ ( = 2&+ ,& • -., -/, … -12/ L
Graph Fourier • ! → #! = % = &', &), … &+,) - GF .- ! = / 0 %010 • U = 1', 1), … 1+,) Graph Fourier #! = 23 ! • Graph Fourier 4#! = 2#!
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 * : ⊙:
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 * : ⊙:
• Graph Convolution (!" = $%&!(()) !" ∗+ ,: = .!"./ , • • • CNN • • U : O(n^2) •
2 Graph Convolutional Network • [2] • [2] [3]
• !" L Λ [2] • K K • O(|E|)
• K=1 [3] ! = #$%& ! ∈ ℝ) × + : % ∈ ℝ) × , : ( ) & ∈ ℝ, × + : #$ ∈ ℝ, × + : GCN
• GCN
• Karate club • 4 • [6]
WordNet • Wang, et al [4] • GCN(Graph Convolutional Network) Zero-shot Learning • Chen, et al [5] • GCN
• Graph Convolution • Graph Fourier Convolution Theorem • Graph Convolution Network • • • GCN
[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]

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[DL Hacks]Graph Convolutional Network LT

  • 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
  • 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]