We show that the classification performance of Graph Convolutional Networks is related to the alignment between features, graph and ground truth, which we quantify using a subspace alignment measure corresponding to the Frobenius norm of the matrix of pairwise chordal distances between three subspaces associated with features, graph and ground truth. The proposed measure is based on the principal angles between subspaces and has both spectral and geometrical interpretations. We showcase the relationship between the subspace alignment measure and the classification performance through the study of limiting cases of Graph Convolutional Networks as well as systematic randomizations of both features and graph structure applied to a constructive example and several examples of citation networks of different origin. The analysis also reveals the relative importance of the graph and features for classification purposes.
The first attempt to generalize neural networks on graphs can be traced back to Gori (2005) @cite_24 , who proposed a scheme combining recurrent neural networks (RNNs) and random walk models. Their method requires the repeated application of contraction maps as propagation functions until the node representations reach a stable fixed point. This method, however, did not attract much attention when it was proposed. With the current surge of interest in deep learning, this work has been reappraised in a new and modern form: Ref. @cite_22 introduced modern techniques for RNN training based on the original graph neural network framework, whereas Ref. @cite_9 proposed a convolution-like propagation rule on graphs and methods for graph-level classification.
@cite_24 @cite_9 @cite_22
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Deep Recurrent Neural Network architectures, though remarkably capable at modeling sequences, lack an intuitive high-level spatio-temporal structure. That is while many problems in computer vision inherently have an underlying high-level structure and can benefit from it. Spatiotemporal graphs are a popular tool for imposing such high-level intuitions in the formulation of real world problems. In this paper, we propose an approach for combining the power of high-level spatio-temporal graphs and sequence learning success of Recurrent Neural Networks (RNNs). We develop a scalable method for casting an arbitrary spatio-temporal graph as a rich RNN mixture that is feedforward, fully differentiable, and jointly trainable. The proposed method is generic and principled as it can be used for transforming any spatio-temporal graph through employing a certain set of well defined steps. The evaluations of the proposed approach on a diverse set of problems, ranging from modeling human motion to object interactions, shows improvement over the state-of-the-art with a large margin. We expect this method to empower new approaches to problem formulation through high-level spatio-temporal graphs and Recurrent Neural Networks. Deep Recurrent Neural Network architectures, though remarkably capable at modeling sequences, lack an intuitive high-level spatio-temporal structure. That is while many problems in computer vision inherently have an underlying high-level structure and can benefit from it. Spatio-temporal graphs are a popular tool for imposing such high-level intuitions in the formulation of real world problems. In this paper, we propose an approach for combining the power of high-level spatio-temporal graphs and sequence learning success of Recurrent Neural Networks (RNNs). We develop a scalable method for casting an arbitrary spatio-temporal graph as a rich RNN mixture that is feedforward, fully differentiable, and jointly trainable. The proposed method is generic and principled as it can be used for transforming any spatio-temporal graph through employing a certain set of well defined steps. The evaluations of the proposed approach on a diverse set of problems, ranging from modeling human motion to object interactions, shows improvement over the state-of-the-art with a large margin. We expect this method to empower new approaches to problem formulation through high-level spatio-temporal graphs and Recurrent Neural Networks. In the first part of this paper, a regular recurrent neural network (RNN) is extended to a bidirectional recurrent neural network (BRNN). The BRNN can be trained without the limitation of using input information just up to a preset future frame. This is accomplished by training it simultaneously in positive and negative time direction. Structure and training procedure of the proposed network are explained. In regression and classification experiments on artificial data, the proposed structure gives better results than other approaches. For real data, classification experiments for phonemes from the TIMIT database show the same tendency. In the second part of this paper, it is shown how the proposed bidirectional structure can be easily modified to allow efficient estimation of the conditional posterior probability of complete symbol sequences without making any explicit assumption about the shape of the distribution. For this part, experiments on real data are reported. In existing convolutional neural networks (CNNs), both convolution and pooling are locally performed for image regions separately, no contextual dependencies between different image regions have been taken into consideration. Such dependencies represent useful spatial structure information in images. Whereas recurrent neural networks (RNNs) are designed for learning contextual dependencies among sequential data by using the recurrent (feedback) connections. In this work, we propose the convolutional recurrent neural network (C-RNN), which learns the spatial dependencies between image regions to enhance the discriminative power of image representation. The C-RNN is trained in an end-to-end manner from raw pixel images. CNN layers are firstly processed to generate middle level features. RNN layer is then learned to encode spatial dependencies. The C-RNN can learn better image representation, especially for images with obvious spatial contextual dependencies. Our method achieves competitive performance on ILSVRC 2012, SUN 397, and MIT indoor.