Generative models for 3D geometric data arise in many important applications in 3D computer vision and graphics. In this paper, we focus on 3D deformable shapes that share a common topological structure, such as human faces and bodies. Morphable Models and their variants, despite their linear formulation, have been widely used for shape representation, while most of the recently proposed nonlinear approaches resort to intermediate representations, such as 3D voxel grids or 2D views. In this work, we introduce a novel graph convolutional operator, acting directly on the 3D mesh, that explicitly models the inductive bias of the fixed underlying graph. This is achieved by enforcing consistent local orderings of the vertices of the graph, through the spiral operator, thus breaking the permutation invariance property that is adopted by all the prior work on Graph Neural Networks. Our operator comes by construction with desirable properties (anisotropic, topology-aware, lightweight, easy-to-optimise), and by using it as a building block for traditional deep generative architectures, we demonstrate state-of-the-art results on a variety of 3D shape datasets compared to the linear Morphable Model and other graph convolutional operators.
a set of recent methods trying to generalize neural network architectures to non-Euclidean domains such as graphs and manifolds @cite_16 . Such methods have achieved very promising results in geometry processing and computer graphics @cite_3 @cite_37 @cite_15 , computational chemistry and drug design @cite_7 @cite_4 , and network science @cite_24 @cite_15 . Multiple approaches have been proposed to construct convolution-like operations on graphs and meshes, including spectral methods @cite_41 @cite_31 @cite_24 @cite_30 , local charting based @cite_3 @cite_19 @cite_21 @cite_13 @cite_25 @cite_27 @cite_43 and graph attention @cite_48 . Finally, in order to aggregate local information, graph or mesh coarsening techniques @cite_31 @cite_9 have been proposed, equivalent to image pooling.
@cite_30 @cite_37 @cite_31 @cite_4 @cite_7 @cite_41 @cite_48 @cite_9 @cite_21 @cite_3 @cite_24 @cite_19 @cite_27 @cite_43 @cite_15 @cite_16 @cite_13 @cite_25
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Two architectures that generalize convolutional neural networks (CNNs) for the processing of signals supported on graphs are introduced. We start with the selection graph neural network (GNN), which replaces linear time invariant filters with linear shift invariant graph filters to generate convolutional features and reinterprets pooling as a possibly nonlinear subsampling stage where nearby nodes pool their information in a set of preselected sample nodes. A key component of the architecture is to remember the position of sampled nodes to permit computation of convolutional features at deeper layers. The second architecture, dubbed aggregation GNN, diffuses the signal through the graph and stores the sequence of diffused components observed by a designated node. This procedure effectively aggregates all components into a stream of information having temporal structure to which the convolution and pooling stages of regular CNNs can be applied. A multinode version of aggregation GNNs is further introduced for operation in large-scale graphs. An important property of selection and aggregation GNNs is that they reduce to conventional CNNs when particularized to time signals reinterpreted as graph signals in a circulant graph. Comparative numerical analyses are performed in a source localization application over synthetic and real-world networks. Performance is also evaluated for an authorship attribution problem and text category classification. Multinode aggregation GNNs are consistently the best-performing GNN architecture. This paper introduces a generalization of Convolutional Neural Networks (CNNs) to graphs with irregular linkage structures, especially heterogeneous graphs with typed nodes and schemas. We propose a novel spatial convolution operation to model the key properties of local connectivity and translation invariance, using high-order connection patterns or motifs. We develop a novel deep architecture Motif-CNN that employs an attention model to combine the features extracted from multiple patterns, thus effectively capturing high-order structural and feature information. Our experiments on semi-supervised node classification on real-world social networks and multiple representative heterogeneous graph datasets indicate significant gains of 6-21 over existing graph CNNs and other state-of-the-art techniques. 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. Deep neural networks have enjoyed remarkable success for various vision tasks, however it remains challenging to apply CNNs to domains lacking a regular underlying structures such as 3D point clouds. Towards this we propose a novel convolutional architecture, termed SpiderCNN, to efficiently extract geometric features from point clouds. SpiderCNN is comprised of units called SpiderConv, which extend convolutional operations from regular grids to irregular point sets that can be embedded in ( R ^n ), by parametrizing a family of convolutional filters. We design the filter as a product of a simple step function that captures local geodesic information and a Taylor polynomial that ensures the expressiveness. SpiderCNN inherits the multi-scale hierarchical architecture from classical CNNs, which allows it to extract semantic deep features. Experiments on ModelNet40 demonstrate that SpiderCNN achieves state-of-the-art accuracy (92.4 ) on standard benchmarks, and shows competitive performance on segmentation task.