Dual Framelets on Manifolds and Graphs with Applications in Multiscale Data Analysis

In recent years, the advance of modern information technologies such as smartphones, AR/VR devices, wearable devices, and so on has changed people's everyday life tremendously. Smartphones provide standard platforms for social activities, e-shopping, entertainment, news, etc.; AR/VR devices provide a new and exciting experiences for interacting and entertaining; wearable digital devices provide individual data acquisition and analysis for personal health and sport activities. In short, we are in a new era of information and behind all these types of information technologies are the generating, storage, processing, transmitting, analyzing, and applications of tremendous size of data. "Data" inevitably becomes the key ingredient of everyone's life. How to effectively make a good use of data plays a crucial role in many aspects of a corporation, a government, a country, and even the whole world.Data collected nowadays are typically high-dimensional, massive in size, and with great complexity. Smart Learning from such a data set for information extraction and applications has become one of the hot-spots of today's scientific research. It is well known that though data are high-dimensional, massive, and with great complexity, they are essentially embedded in some manifolds or can be represented as graphs, which are structurally low-dimensional and sparse that can be exploited by multiscale representation systems such as wavelets, framelets, etc. Dual framelets is one of the most important types of multiscale representation systems due to their great variety in design and high flexibility in applications.In this project, we shall focus on the characterizations and construction of dual framelets on compact Riemannian manifolds and graphs (or networks), and their applications in multiscale data analysis. We shall focus on the tasks of investigating sequences of dual framelets, studying their MRA structure and filter bank connections, providing the construction of solid examples of dual framelets on manifolds such as dual framelets on sphere, torus, Grassmannian, as well as on graphs such as graphs from machine learning related to social networks, neural networks, etc. We shall also investigate in practice the fast dual framelet transforms on manifolds and graphs and their applications. The results of our project will not only provide theoretically important characterizations of multiscale representation systems on general domains including manifolds and graphs, but also practically useful transform algorithms that are fast and efficient for multiscale data analysis.