Framelets for Geometric Deep Learning: Spheres, Graphs, and Neural Networks

Deep learning based on the architecture of multilayer artificial neural networks has revolutionized modern society and people's daily life during the past decade ranging from automated driving, online shopping, AI assistant, to smart surveillant systems, medical diagnosis, drug discovery, and so on. While deep learning models, such as the convolutional neural networks, have been successfully used in speech recognition, image classification, or video processing, where the underlying domain of data is the Euclidean space, geometric deep learning goes beyond the training and learning of data defined on Euclidian domains and deal with commonly appeared non-Euclidean structured data in real-life applications, such as graph data and manifold data. For example, in network sciences, the road network dataset includes traffic directions and road junctions are (directed) graph data that can be used for supporting the development of intelligent transport system; in biomedical and drug industry, the drug–target–indication interaction and relationship provide a graph-structured data; in geology, remote sensing data are naturally spherical signals; and in computer vision, 3D objects are modeled as Riemannian surfaces with color texture. Geometric deep learning aims at encoding geometric priors of non-Euclidean data into the design of deep neural network models to improve the overall neural network performance. Framelet systems, naturally equipped with the properties of multiscale analysis, sparse representation, customized filter bank design, fast algorithmic implementation, and so on, provide a powerful mathematical tool for geometric deep learning. In this project, we focus on two of the most important mathematical objects, the graphs and the spheres, and consider (1) the development of framelet systems on spheres and graphs with desirable properties; (2) the integration of framelet systems into the design of neural network models for geometric deep learning; (3) the theoretical connections between the framelet systems as well as the neural network models from the viewpoints of discretization and continuity; and (4) the extension and generalization of the framelet systems and their respected neural network models to more general domains. The outcomes of this project would be powerful framelet systems and deep learning models with rich mathematical theory for geometric deep learning.