Geometric deep learning: going beyond Euclidean data

Many signal processing problems involve data whose underlying structure is non-Euclidean, but may be modeled as a manifold or (combinatorial) graph. For instance, in social networks, the characteristics of users can be modeled as signals on the vertices of the social graph [1]. Sensor networks are graph models of distributed interconnected sensors, whose readings are modelled as time-dependent signals on the vertices. In genetics, gene expression data are modeled as signals defined on the regulatory network [2]. In neuroscience, graph models are used to represent anatomical and functional structures of the brain. In computer graphics and vision, 3D objects are modeled as Riemannian manifolds (surfaces) endowed with properties such as color texture. Even more complex examples include networks of operators, e.g., functional correspondences [3] or difference operators [4] in a collection of 3D shapes, or orientations of overlapping cameras in multi-view vision (“structure from motion”) problems [5]. The complexity of geometric data and the availability of very large datasets (in the case of social networks, on the scale of billions) suggest the use of machine learning techniques. In particular, deep learning has recently proven to be a powerful tool for problems with large datasets with underlying Euclidean structure. The purpose of this paper is to overview the problems arising in relation to geometric deep learning and present solutions existing today for this class of problems, as well as key difficulties and future research directions.