We introduce an architecture based on deep hierarchical decompositions to learn effective representations of large graphs. Our framework extends classic R-decompositions used in kernel methods, enabling nested "part-of-part" relations. Unlike recursive neural networks, which unroll a template on input graphs directly, we unroll a neural network template over the decomposition hierarchy, allowing us to deal with the high degree variability that typically characterize social network graphs. Deep hierarchical decompositions are also amenable to domain compression, a technique that reduces both space and time complexity by exploiting symmetries. We show empirically that our approach is competitive with current state-of-the-art graph classification methods, particularly when dealing with social network datasets.
@math @cite_24 casts graphs into a format suitable for learning convolutional neural networks ( @math s): 1) graphs are decomposed into a fixed number of neighborhood subgraphs; 2) which are then casted to a fixed-size receptive field. Both 1) and 2) involve either padding or truncation in order to meet the fixed-size requirements. The truncation operation can be detrimental for the statistical performance of the downstream @math since it throws away part of the input graph. On the other hand @math is able to handle structured inputs of variable sizes without throwing away part of the them. And this is one of the reasons because @math has better statistical performance than @math (See ).
@cite_24
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Conventional convolutional neural networks use either a linear or a nonlinear filter to extract features from an image patch (region) of spatial size @math (typically, @math is small and is equal to @math , e.g., @math is 5 or 7). Generally, the size of the filter is equal to the size @math of the input patch. We argue that the representational ability of equal-size strategy is not strong enough. To overcome the drawback, we propose to use subpatch filter whose spatial size @math is smaller than @math . The proposed subpatch filter consists of two subsequent filters. The first one is a linear filter of spatial size @math and is aimed at extracting features from spatial domain. The second one is of spatial size @math and is used for strengthening the connection between different input feature channels and for reducing the number of parameters. The subpatch filter convolves with the input patch and the resulting network is called a subpatch network. Taking the output of one subpatch network as input, we further repeat constructing subpatch networks until the output contains only one neuron in spatial domain. These subpatch networks form a new network called the cascaded subpatch network (CSNet). The feature layer generated by CSNet is called the csconv layer. For the whole input image, we construct a deep neural network by stacking a sequence of csconv layers. Experimental results on five benchmark data sets demonstrate the effectiveness and compactness of the proposed CSNet. For example, our CSNet reaches a test error of 5.68 on the CIFAR10 data set without model averaging. To the best of our knowledge, this is the best result ever obtained on the CIFAR10 data set.