Graph Based Convolutional Neural Network

The benefit of localized features within the regular domain has given rise to the use of Convolutional Neural Networks (CNNs) in machine learning, with great proficiency in the image classification. The use of CNNs becomes problematic within the irregular spatial domain due to design and convolution of a kernel filter being non-trivial. One solution to this problem is to utilize graph signal processing techniques and the convolution theorem to perform convolutions on the graph of the irregular domain to obtain feature map responses to learnt filters. We propose graph convolution and pooling operators analogous to those in the regular domain. We also provide gradient calculations on the input data and spectral filters, which allow for the deep learning of an irregular spatial domain problem. Signal filters take the form of spectral multipliers, applying convolution in the graph spectral domain. Applying smooth multipliers results in localized convolutions in the spatial domain, with smoother multipliers providing sharper feature maps. Algebraic Multigrid is presented as a graph pooling method, reducing the resolution of the graph through agglomeration of nodes between layers of the network. Evaluation of performance on the MNIST digit classification problem in both the regular and irregular domain is presented, with comparison drawn to standard CNN. The proposed graph CNN provides a deep learning method for the irregular domains present in the machine learning community, obtaining 94.23% on the regular grid, and 94.96% on a spatially irregular subsampled MNIST.