a model to recover topology can enrich the representation power of point clouds. To this end, we propose a new neural network module dubbed Edge-Conv 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.

Many scientific fields study data with an underlying structure that is a non-Euclidean space. 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 is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field.

Deep learning has achieved a remarkable performance breakthrough in several fields, most notably in speech recognition, natural language processing, and computer vision. In particular, convolutional neural network (CNN) architectures currently produce state-of-the-art performance on a variety of image analysis tasks such as object detection and recognition. Most of deep learning research has so far focused on dealing with 1D, 2D, or 3D Euclideanstructured data such as acoustic signals, images, or videos. Recently, there has been an increasing interest in geometric deep learning, attempting to generalize deep learning methods to non-Euclidean structured data such as graphs and manifolds, with a variety of applications from the domains of network analysis, computational social science, or computer graphics. In this paper, we propose a unified framework allowing to generalize CNN architectures to non-Euclidean domains (graphs and manifolds) and learn local, stationary, and compositional task-specific features. We show that various non-Euclidean CNN methods previously proposed in the literature can be considered as particular instances of our framework. We test the proposed method on standard tasks from the realms of image-, graphand 3D shape analysis and show that it consistently outperforms previous approaches.

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Feature descriptors play a crucial role in a wide range of geometry analysis and processing applications, including shape correspondence, retrieval, and segmentation. In this paper, we introduce Geodesic Convolutional Neural Networks (GCNN), a generalization of the convolutional networks (CNN) paradigm to non-Euclidean manifolds. Our construction is based on a local geodesic system of polar coordinates to extract "patches", which are then passed through a cascade of filters and linear and non-linear operators. The coefficients of the filters and linear combination weights are optimization variables that are learned to minimize a task-specific cost function. We use GCNN to learn invariant shape features, allowing to achieve state-of-the-art performance in problems such as shape description, retrieval, and correspondence.

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An efficient algorithm for isometry-invariant matching of surfaces is presented. The key idea is computing the minimum-distortion mapping between two surfaces. For this purpose, we introduce the generalized multidimensional scaling, a computationally efficient continuous optimization algorithm for finding the least distortion embedding of one surface into another. The generalized multidimensional scaling algorithm allows for both full and partial surface matching. As an example, it is applied to the problem of expression-invariant three-dimensional face recognition.Gromov-Hausdorff distance ͉ isometric embedding ͉ iterative-closestpoint ͉ partial embedding T he problem of comparison between deformable surfaces has recently gained a lot of interest in pattern recognition, especially in the fields of three-dimensional (3D) data analysis and synthesis. It arises, for example, in medical imaging and 3D face recognition (1). The fundamental question is how to efficiently yet accurately quantify the similarity between a given reference surface ("model") and some other surface ("probe"), usually a bent version of the model. We would like to capture the distinction of the model and the probe intrinsic properties associated with the metric structure of the surface, while ignoring the extrinsic properties that describe the way the surface is immersed into the ambient space and that often change while the surface bends. A deformation that preserves the intrinsic structure of the surface is called an ''isometry.'' Our goal is thus to define a computable isometry-invariant measure of similarity between surfaces. Often, the problem is even more complicated, when having the probe only partially available (see Fig. 1). We refer to the latter setting as ''partial surface matching'' and to the case where the whole probe is available as ''full surface matching.'' One of the earliest attempts of isometry-invariant surface matching is the classical ''iterative closest point'' (ICP) algorithm (2). It addresses a particular case of the partial matching problem where only rigid (Euclidean) isometries are allowed. An efficient method for the construction of near-isometry-invariant representations of surfaces [called ''canonical forms'' (CFs)] based on Euclidean embeddings was presented in ref. 3, as a generalization of ref. 4. This approach used a multidimensional scaling (MDS) algorithm (5). MDS is closely related to dimensionality reduction (6, 7) and can be performed in a computationally efficient manner. Euclidean embeddings are used in theoretical computer science for representing metric spaces usually arising from geometry of graphs (8).Recently, Mémoli and Sapiro (9) used a discrete probabilistic approximation of the Gromov-Hausdorff (GH) distance (10) to compare surfaces up to isometries. Their method is based on evaluating all of the permutations between two sets of surface samples, which is computationally expensive. Moreover, the GH distance is inappropriate for partial matching.One of the main contributions of this work is t...