Convolutional Neural Networks on Graphs

with Fast Localized Spectral FilteringDefferrard, Michaël, Xavier Bresson, and Pierre Van-

dergheynstNIPS 2016

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, tele-

phony.

• Graphs can model heterogeneous pairwise relation-ships.• Graphs can encode complex geometric structures.

CNN architecture

• Convolution filter translation or fast Fourier transform (FFT).• Down-sampling pick one pixel out of n.

Generalizing CNNs to graphs• Challenges• Formulate convolution and down-sam-

pling on graphs• How to define localized graph filters?• Make them efficient

Generalizing CNNs to graphs1. The design of localized convolutional filters on graphs2. Graph coarsening procedure (sub-sampling)3. Graph pooling operation

• : undirected and con-nected graph

• Spectral graph theory• Graph Laplacians

• Normalized Laplacians

: set of vertices : set of edges : weighted adjacency matrix : diagonal degree matrix : identity matrix

Graph Fourier Transform

Graph Fourier Transform• Graph Fourier Transform• (Eigen value decomposition)• Graph Fourier basis • Graph frequencies = 1. Graph signal , 2. Transform

Spectral filtering of graph signals• Convolution on graphs

• filtered signal

• A non-parametric filter Non-localized in vertex domain Learning complexity in Computational complexity in

Polynomial parametrization for localized fil-ters•

order polynomials of the Laplacian -> -localized Learning complexity in Still, computational complexity in because of multiplication with Fourier

basis

• Filter localization on graph

Recursive formulation for fast filter-ing

• Chebyshev expansion • Filtered • multiplications by a sparse costs

Learning complexity in Computational complexity in

Graph coarsening and pooling

• Graph coarsening• To cluster similar vertices together, multilevel clustering algorithm is needed.• Pick an unmarked vertex and matching it with one of its unmarked neighbors that maxi-

mizes the local normalized cut • Pooling of graph signals

• Balanced binary tree structured coarsened graphs• ReLU activation with max pooling

• e.g.

level 0

level 1

level 2

Graph ConvNet (GCN) architecture

Experiments• MNIST

• CNNs on a Euclidean space• Comparable to classical CNN• Isotropic spectral filters

• edges in a general graph do not pos-sess an orientation

Experiments• 20NEWS

• structure documents with a feature graph

• 10,000 nodes, 132,834 edges

𝑂 (𝑛2)

𝑂 (𝑛)

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

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/)