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.
We now present briefly the key insights introduced by Bruna @cite_26 to extend CNNs to the non-Euclidean domain. For an extensive recent review, the reader should refer to @cite_18 .
@cite_18 @cite_26
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Convolutional neural networks (CNNs) have achieved great successes in many computer vision problems. Unlike existing works that designed CNN architectures to improve performance on a single task of a single domain and not generalizable, we present IBN-Net, a novel convolutional architecture, which remarkably enhances a CNN’s modeling ability on one domain (e.g. Cityscapes) as well as its generalization capacity on another domain (e.g. GTA5) without finetuning. IBN-Net carefully integrates Instance Normalization (IN) and Batch Normalization (BN) as building blocks, and can be wrapped into many advanced deep networks to improve their performances. This work has three key contributions. (1) By delving into IN and BN, we disclose that IN learns features that are invariant to appearance changes, such as colors, styles, and virtuality reality, while BN is essential for preserving content related information. (2) IBN-Net can be applied to many advanced deep architectures, such as DenseNet, ResNet, ResNeXt, and SENet, and consistently improve their performance without increasing computational cost. (3) When applying the trained networks to new domains, e.g. from GTA5 to Cityscapes, IBN-Net achieves comparable improvements as domain adaptation methods, even without using data from the target domain. With IBN-Net, we won the 1st place on the WAD 2018 Challenge Drivable Area track, with an mIoU of 86.18 . Recently, very deep convolutional neural networks (CNNs) have shown outstanding performance in object recognition and have also been the first choice for dense classification problems such as semantic segmentation. However, repeated subsampling operations like pooling or convolution striding in deep CNNs lead to a significant decrease in the initial image resolution. Here, we present RefineNet, a generic multi-path refinement network that explicitly exploits all the information available along the down-sampling process to enable high-resolution prediction using long-range residual connections. In this way, the deeper layers that capture high-level semantic features can be directly refined using fine-grained features from earlier convolutions. The individual components of RefineNet employ residual connections following the identity mapping mindset, which allows for effective end-to-end training. Further, we introduce chained residual pooling, which captures rich background context in an efficient manner. We carry out comprehensive experiments and set new state-of-the-art results on seven public datasets. In particular, we achieve an intersection-over-union score of 83.4 on the challenging PASCAL VOC 2012 dataset, which is the best reported result to date. Recently, very deep convolutional neural networks (CNNs) have shown outstanding performance in object recognition and have also been the first choice for dense classification problems such as semantic segmentation. However, repeated subsampling operations like pooling or convolution striding in deep CNNs lead to a significant decrease in the initial image resolution. Here, we present RefineNet, a generic multi-path refinement network that explicitly exploits all the information available along the down-sampling process to enable high-resolution prediction using long-range residual connections. In this way, the deeper layers that capture high-level semantic features can be directly refined using fine-grained features from earlier convolutions. The individual components of RefineNet employ residual connections following the identity mapping mindset, which allows for effective end-to-end training. Further, we introduce chained residual pooling, which captures rich background context in an efficient manner. We carry out comprehensive experiments and set new state-of-the-art results on seven public datasets. In particular, we achieve an intersection-over-union score of 83.4 on the challenging PASCAL VOC 2012 dataset, which is the best reported result to date. We propose a generalization of convolutional neural networks (CNNs) to irregular domains, through the use of an inferred graph structure. In more details, we introduce a three-step methodology to create convolutional layers that are adapted to the signals to process: 1) From a training set of signals, infer a graph representing the topology on which they evolve; 2) Identify translation operators in the vertex domain; 3) Emulate a convolution operator by translating a localized kernel on the graph. Using these layers, a convolutional neural network is built, and is trained on the initial signals to perform a classification task. Contributions are twofold. First, we adapt a definition of translations on graphs to make them more robust to irregularities, and to take into account locality of the kernel. Second, we introduce a procedure to build CNNs from data. We apply our methodology on a scrambled version of the CIFAR-10 and Haxby datasets. Without using any knowledge on the signals, we significantly outperform existing methods. Moreover, our approach extends classical CNNs on images in the sense that such networks are a particular case of our approach when the inferred graph is a grid.