Convolutional neural networks (CNNs) have greatly improved state-of-the-art performances in a number of fields, notably computer vision and natural language processing. In this work, we are interested in generalizing the formulation of CNNs from low-dimensional regular Euclidean domains, where images (2D), videos (3D) and audios (1D) are represented, to high-dimensional irregular domains such as social networks or biological networks represented by graphs. This paper introduces a formulation of CNNs on graphs in the context of spectral graph theory. We borrow the fundamental tools from the emerging field of signal processing on graphs, which provides the necessary mathematical background and efficient numerical schemes to design localized graph filters efficient to learn and evaluate. As a matter of fact, we introduce the first technique that offers the same computational complexity than standard CNNs, while being universal to any graph structure. Numerical experiments on MNIST and 20NEWS demonstrate the ability of this novel deep learning system to learn local, stationary, and compositional features on graphs, as long as the graph is well-constructed. arXiv:1606.09375v1 [cs.LG] 30 Jun 2016 1. Spectral formulation. A graph spectral theoretical formulation of CNNs on graphs built on established tools in GSP. 2. Strictly localized filters. Enhancing [4, 15] , the proposed spectral filters are provable to be strictly localized in a ball of radius K, i.e. K hops from the central vertex. 3. Low computational complexity. The evaluation complexity of our filters is linear w.r.t. the filters support's size K and the number of edges |E|. Importantly, as most real-world graphs are highly sparse, we have |E| n 2 and |E| = kn for the widespread k-nearest neighbor (NN) graphs, leading to a linear complexity w.r.t the input data size. Moreover, this method avoids the Fourier basis altogether, thus the expensive eigenvalue decomposition (EVD) necessary to
In REF , the spectral graph theoretical of CNNs on graphs is formulated and strictly localized spectral filters are given.
Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering
Advances in Neural Information Processing Systems 29 (2016)
Computer Science Mathematics