Deep learning has revolutionized many machine learning tasks in recent years, ranging from image classification and video processing to speech recognition and natural language understanding. The data in these tasks are typically represented in the Euclidean space. However, there is an increasing number of applications, where data are generated from non-Euclidean domains and are represented as graphs with complex relationships and interdependency between objects. The complexity of graph data has imposed significant challenges on the existing machine learning algorithms. Recently, many studies on extending deep learning approaches for graph data have emerged. In this article, we provide a comprehensive overview of graph neural networks (GNNs) in data mining and machine learning fields. We propose a new taxonomy to divide the state-of-the-art GNNs into four categories, namely, recurrent GNNs, convolutional GNNs, graph autoencoders, and spatial–temporal GNNs. We further discuss the applications of GNNs across various domains and summarize the open-source codes, benchmark data sets, and model evaluation of GNNs. Finally, we propose potential research directions in this rapidly growing field.

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A Comprehensive Survey on Graph Neural Networks

Point clouds provide a flexible geometric representation suitable for countless applications in computer graphics; they also comprise the raw output of most 3D data acquisition devices. While hand-designed features on point clouds have long been proposed in graphics and vision, however, the recent overwhelming success of convolutional neural networks (CNNs) for image analysis suggests the value of adapting insight from CNN to the point cloud world. Point clouds inherently lack topological information, so designing a model to recover topology can enrich the representation power of point clouds. To this end, we propose a new neural network module dubbed EdgeConv suitable for CNN-based high-level tasks on point clouds, including classification and segmentation. EdgeConv acts on graphs dynamically computed in each layer of the network. It is differentiable and can be plugged into existing architectures. Compared to existing modules operating in extrinsic space or treating each point independently, EdgeConv has several appealing properties: It incorporates local neighborhood information; it can be stacked applied to learn global shape properties; and in multi-layer systems affinity in feature space captures semantic characteristics over potentially long distances in the original embedding. We show the performance of our model on standard benchmarks, including ModelNet40, ShapeNetPart, and S3DIS.

/pdf/dynamic-graph-cnn-for-learning-on-point-clouds-20wlcru9tg.pdf

Dynamic Graph CNN for Learning on Point Clouds

Many scientific fields study data with an underlying structure that is non-Euclidean. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions) and are natural targets for machine-learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural-language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure and in cases where the invariances of these structures are built into networks used to model them.

Geometric Deep Learning: Going beyond Euclidean data

Lots of learning tasks require dealing with graph data which contains rich relation information among elements. Modeling physics systems, learning molecular fingerprints, predicting protein interface, and classifying diseases demand a model to learn from graph inputs. In other domains such as learning from non-structural data like texts and images, reasoning on extracted structures (like the dependency trees of sentences and the scene graphs of images) is an important research topic which also needs graph reasoning models. Graph neural networks (GNNs) are neural models that capture the dependence of graphs via message passing between the nodes of graphs. In recent years, variants of GNNs such as graph convolutional network (GCN), graph attention network (GAT), graph recurrent network (GRN) have demonstrated ground-breaking performances on many deep learning tasks. In this survey, we propose a general design pipeline for GNN models and discuss the variants of each component, systematically categorize the applications, and propose four open problems for future research.

Graph Neural Networks: A Review of Methods and Applications

In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.

Evolution and the Theory of Games

Quantile regression (QR) is a statistical tool for distribution-free estimation of conditional quantiles of a target variable given explanatory features. QR is limited by the assumption that the target distribution is univariate and defined on an Euclidean domain. Although the notion of quantiles was recently extended to multi-variate distributions, QR for multi-variate distributions on manifolds remains underexplored, even though many important applications inherently involve data distributed on, e.g., spheres (climate measurements), tori (dihedral angles in proteins), or Lie groups (attitude in navigation). By leveraging optimal transport theory and the notion of $c$-concave functions, we meaningfully define conditional vector quantile functions of high-dimensional variables on manifolds (M-CVQFs). Our approach allows for quantile estimation, regression, and computation of conditional confidence sets. We demonstrate the approach's efficacy and provide insights regarding the meaning of non-Euclidean quantiles through preliminary synthetic data experiments.

Emanuele Rodolà

Papers

Vector Quantile Regression on Manifolds