While convolutional neural networks (CNNs) have recently made great strides in supervised classification of data structured on a grid (e.g. images composed of pixel grids), in several interesting datasets, the relations between features can be better represented as a general graph instead of a regular grid. Although recent algorithms that adapt CNNs to graphs have shown promising results, they mostly neglect learning explicit operations for edge features while focusing on vertex features alone. We propose new formulations for convolutional, pooling, and fully connected layers for neural networks that make more comprehensive use of the information available in multi-dimensional graphs. Using these layers led to an improvement in classification accuracy over the state-of-the-art methods on benchmark graph datasets.
Spatial approaches do not rely on spectral graph theory to perform filtering. Instead, different architectures define different ways to learn to combine the information from neighboring vertices in the form of local filters. For example, PATCHYSAN transforms the graph data into a grid structure using graph normalization schemes to sort the vertices @cite_10 . This grid structure can be then passed to a traditional CNN. DeepWalk uses random walks to perform localized learning over the neighborhood @cite_7 . SplineCNN learns a spline function using the B-spline basis to sample convolutional weights based on the relative pseudo-coordinates of the neighboring vertices @cite_0 .
@cite_0 @cite_10 @cite_7
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We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant of deep neural networks for irregular structured and geometric input, e.g., graphs or meshes. Our main contribution is a novel convolution operator based on B-splines, that makes the computation time independent from the kernel size due to the local support property of the B-spline basis functions. As a result, we obtain a generalization of the traditional CNN convolution operator by using continuous kernel functions parametrized by a fixed number of trainable weights. In contrast to related approaches that filter in the spectral domain, the proposed method aggregates features purely in the spatial domain. In addition, SplineCNN allows entire end-to-end training of deep architectures, using only the geometric structure as input, instead of handcrafted feature descriptors. For validation, we apply our method on tasks from the fields of image graph classification, shape correspondence and graph node classification, and show that it outperforms or pars state-of-the-art approaches while being significantly faster and having favorable properties like domain-independence. Numerous important problems can be framed as learning from graph data. We propose a framework for learning convolutional neural networks for arbitrary graphs. These graphs may be undirected, directed, and with both discrete and continuous node and edge attributes. Analogous to image-based convolutional networks that operate on locally connected regions of the input, we present a general approach to extracting locally connected regions from graphs. Using established benchmark data sets, we demonstrate that the learned feature representations are competitive with state of the art graph kernels and that their computation is highly efficient. We present DeepWalk, a novel approach for learning latent representations of vertices in a network. These latent representations encode social relations in a continuous vector space, which is easily exploited by statistical models. DeepWalk generalizes recent advancements in language modeling and unsupervised feature learning (or deep learning) from sequences of words to graphs. DeepWalk uses local information obtained from truncated random walks to learn latent representations by treating walks as the equivalent of sentences. We demonstrate DeepWalk's latent representations on several multi-label network classification tasks for social networks such as BlogCatalog, Flickr, and YouTube. Our results show that DeepWalk outperforms challenging baselines which are allowed a global view of the network, especially in the presence of missing information. DeepWalk's representations can provide F1 scores up to 10 higher than competing methods when labeled data is sparse. In some experiments, DeepWalk's representations are able to outperform all baseline methods while using 60 less training data. DeepWalk is also scalable. It is an online learning algorithm which builds useful incremental results, and is trivially parallelizable. These qualities make it suitable for a broad class of real world applications such as network classification, and anomaly detection.