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Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering

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Paper review
"Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering", NIPS 2016

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Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering

  1. 1. Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering Defferrard, Michaël, Xavier Bresson, and Pierre Vandergheynst NIPS 2016
  2. 2. Unstructured data as graphs • Majority of data is naturally unstructured, but can be structured. • Irregular / non-Euclidean data can be structured with graphs • Social networks: Facebook, Twitter. • Biological networks: genes, molecules, brain connectivity. • Infrastructure networks: energy, transportation, Internet, telephony. • Graphs can model heterogeneous pairwise relationships. • Graphs can encode complex geometric structures.
  3. 3. CNN architecture • Convolution filter translation or fast Fourier transform (FFT). • Down-sampling pick one pixel out of n.
  4. 4. Generalizing CNNs to graphs • Challenges • Formulate convolution and down-sampling on graphs • How to define localized graph filters? • Make them efficient
  5. 5. Generalizing CNNs to graphs 1. The design of localized convolutional filters on graphs 2. Graph coarsening procedure (sub-sampling) 3. Graph pooling operation
  6. 6. • 𝐺 = (𝑉, 𝐸, 𝑊) : undirected and connected graph • Spectral graph theory • Graph Laplacians • 𝐿 = 𝐷 − 𝑊 • Normalized Laplacians 𝐿 = 𝐼 𝑛 – 𝐷− 1 2 𝑊𝐷− 1 2  𝑉 : set of vertices  𝐸 : set of edges  𝑊 : weighted adjacency matrix  𝐷𝑖𝑖 = 𝑗 𝑊𝑖𝑗 : diagonal degree matrix  𝐼 𝑛 : identity matrix Graph Fourier Transform
  7. 7. Graph Fourier Transform • Graph Fourier Transform • 𝐿 = 𝑈Λ𝑈 𝑇 (Eigen value decomposition) • Graph Fourier basis 𝑈 = [𝑢0, … , 𝑢 𝑛−1] • Graph frequencies Λ = 𝜆0 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 𝜆 𝑛−1 1. Graph signal 𝑥 ∶ 𝑉 → ℝ, 𝑥 ∈ ℝ 𝑛 2. Transform 𝑥 = 𝑈 𝑇 𝑥 ∈ ℝ 𝑛
  8. 8. Spectral filtering of graph signals • Convolution on graphs • 𝑥 ∗ 𝒢 𝑦 = 𝑈 𝑈 𝑇 𝑥 ⊙ 𝑈 𝑇 𝑦 • filtered signal 𝑦 = 𝑔 𝜃 L x = 𝑔 𝜃 UΛ𝑈 𝑇 x = 𝑈𝑔 𝜃 Λ 𝑈 𝑇 𝑥 = 𝑈 𝑔 𝜃(𝜆0) ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 𝑔 𝜃(𝜆 𝑛−1) 𝑈 𝑇 𝑥 • A non-parametric filter 𝑔 𝜃 Λ = diag 𝜃 , 𝜃 ∈ ℝ 𝑛  Non-localized in vertex domain  Learning complexity in O(n)  Computational complexity in O(n2)
  9. 9. Polynomial parametrization for localized filters • 𝑔 𝜃 Λ = diag 𝜃 , 𝜃 ∈ ℝ 𝑛 𝑔 𝜃 Λ = 𝑘=0 𝐾−1 𝜃 𝑘Λ 𝑘 , 𝜃 ∈ ℝ 𝑘  𝐾 𝑡ℎ order polynomials of the Laplacian -> 𝐾-localized  Learning complexity in O(K)  Still, computational complexity in O(n2 ) because of multiplication with Fourier basis U • Filter localization on graph
  10. 10. Recursive formulation for fast filtering • 𝑔 𝜃 Λ = 𝑘=0 𝐾−1 𝜃 𝑘Λ 𝑘 , 𝜃 ∈ ℝ 𝑘 𝑔 𝜃 Λ = 𝑘=0 𝐾−1 𝜃 𝑘Tk(Λ) , 𝜃 ∈ ℝ 𝑘 • Chebyshev expansion 𝑇𝑘(𝑥) = 2𝑥𝑇𝑘−1(𝑥) − 𝑇𝑘−2(𝑥) • Filtered 𝑦 = 𝑔 𝜃 𝐿 𝑥 • 𝐾 multiplications by a sparse 𝐿 costs 𝑂 𝐾 𝐸 ≪ 𝑂(𝑛2)  Learning complexity in 𝑂(𝐾)  Computational complexity in 𝑂(𝐾|𝐸|)
  11. 11. Graph coarsening and pooling • Graph coarsening • To cluster similar vertices together, multilevel clustering algorithm is needed. • Pick an unmarked vertex 𝑖 and matching it with one of its unmarked neighbors 𝑗 that maximizes the local normalized cut 𝑊𝑖𝑗( 1 𝑑 𝑖 + 1 𝑑 𝑗 ) • Pooling of graph signals • Balanced binary tree structured coarsened graphs • ReLU activation with max pooling • e.g. 𝑧 = max 𝑥0, 𝑥1 , max 𝑥4, 𝑥5, 𝑥6 , max 𝑥8, 𝑥9, 𝑥10 ∈ ℝ3 level 0 level 1 level 2
  12. 12. Graph ConvNet (GCN) architecture
  13. 13. Experiments • MNIST • CNNs on a Euclidean space • Comparable to classical CNN • Isotropic spectral filters • edges in a general graph do not possess an orientation
  14. 14. Experiments • 20NEWS • structure documents with a feature graph • 10,000 nodes, 132,834 edges 𝑂(𝑛2 ) 𝑂(𝑛)
  15. 15. Conclusion • Contributions • Spectral formulation of CNNs on graphs in GSP • Strictly localized spectral filters are proposed • Linear complexity of filters • Efficient pooling on graphs • Limitation • Filters are not directly transferrable to a different graph
  16. 16. References • Deep Learning on Graphs, a lecture on A Network Tour of Data Science (NTDS) 2016 • Shuman, David I., et al. "The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains." IEEE Signal Processing Magazine 30.3 (2013): 83-98. • How powerful are Graph Convolutions? (http://www.inference.vc/how- powerful-are-graph-convolutions-review-of-kipf-welling-2016-2/) • GRAPH CONVOLUTIONAL NETWORKS (http://tkipf.github.io/graph- convolutional-networks/)

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