  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.
