Directional Framelets on Compact Sets: Theory, Construction, Realization, and Applications
DescriptionThe recent rapid development and advancement in deep/machine learning based on multilayer artificial neural networks (ANNs) has had a huge impact on people's everyday life. Many AI (artificial intelligence) technologies involving deep neural networks are used in smartphones, smart TVs, automated driving, online shopping, surveillant systems, medical diagnosis, and so on. For example, our daily shopping online (e.g., Amazon) could have been taken care of by a pre-trained underlying recommender system utilizing our historical product preferences; the photos we took in smartphones could have been pre-processed by a digital signal processor (DSP) integrated with tremendous AI-related algorithms; the cars we drive could have been installed with an auto-pilot system allowing us to enjoy a “hand-free” long travelling; and so on so forth. The key that deep/machine learning is so popular and successful lies in three aspects: data, algorithms, and the computing power of modern hardware systems. The computing power of modern hardware systems follows the “Moore’s Law”, which grows exponentially and plays the role of the “heart” as in a human body. Then, the algorithms behind all those AI technologies developed by scientists including mathematicians, engineers, programmers, artists, etc., is of course the “brain”, and the data available (collected) for training and testing the AI system is the “food and nutrition” that shape the “muscles and bones” of it, which significantly affects the successfulness of the AI systems.Directional framelet systems have been a powerful tool in the classical data-processing tasks including signal/image processing, computer graphics, numerical PDEs, etc. The rapid advancement of deep /machine learning for the tasks of big data processing not only sheds light on new directions but also raises new challenges on the development of directional framelet systems in both theory and applications. In this project, we shall focus on the development of directional framelet systems on compact sets, including but not restricted to intervals, compact Riemannian manifolds, graphs/networks, etc. Moreover, we shall consider their algorithmic realizations and applications in deep/machine learning related tasks. The data in deep/machine learning varies from tasks to tasks including signals, images, videos, graphs, strings, texts, and so on. We use compact sets to provide a unified mathematical description of data for our further study and investigation. Our investigation of this project would be new perspectives on directional framelet systems with desirable properties, novel representation systems for general data processing on compact sets, as well as deep understanding of sparse representation properties of framelet systems on related domains. Furthermore, our project will provide efficient and effective algorithmic realizations of a large family of directional framelet systems on commonly used domains, significant improvement in the performance of deep/machine learning related tasks, and the cutting-edge techniques for traditional and modern deep/machine learning tasks.
|Effective start/end date||1/01/23 → …|