LEAPS-MPS: Uncovering and Exploiting Multiscale Structures in Big Data Using Diffusion-Based Representation and Optimal Sampling

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Consumers of information desire efficient methods for processing data, and scientists are interested in understanding patterns and structures in data to explain why various phenomena occur. The focus of emerging big data analytics research has shifted to alternative data structures, especially (combinatorial) graphs, and networks. Although graphs and networks require mathematically complex algorithms, they have built-in links that highlight important relationships between nodes and make them simpler to interpret in applications. This project introduces new theories and algorithms to address aspects of graph data challenges that have impacts on diverse segments of society including imaging tools used by medical professionals to diagnose and treat diseases, and on applications ranging from the regulatory networks describing the interactions between genes, RNA, and proteins in the brain, to hardware and sensor technology. The methods explored expand the family of harmonic analysis algorithms that produced tools such as the wavelet-based ones for images used in commercial applications and sampling theorems for telegraph communication. The project also involves training and mentoring undergraduate students, particularly students from underrepresented groups with less exposure to careers in science and technology. The intent is to motivate and prepare them for careers in advanced mathematical sciences and increase the global competitiveness of the US-based STEM workforce. Other activities include promoting equity and inclusion in STEM fields through public lectures and coordinated outreach activities. Applied harmonic analysis tools will play a central role in the two-year project. The primary research objectives include: (i) enhancing the quality of multiscale representations of data signals on graphs and networks, (ii) the optimization of sampling strategies for high-dimensional, non-bandlimited signals, and (iii) the design of compression algorithms to demonstrate the potential for datasets on graphs. The analysis combines frame theory, diffusion geometry, and directional multiscale modeling; novel numerical techniques will be developed using numerical linear algebra and computational harmonic analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.