CIF:Small:Theoretical Foundations for Robust Signal Processing on Spatial Networks

Many complex and interconnected phenomena in the world - such as social media, sensor grids, and brain connectivity - can be modeled using graphs or networks. Unlike classical signal processing, which works with data regularly arranged in a homogeneous space (like audio or images), graph signal processing (GSP) analyzes signals that lie on irregular graphs or networks. Important outcomes of such analysis include detecting patterns, detecting and reducing noise, and visualizing the network. GSP has become a vibrant field of research in engineering and mathematics due to its applicability to a wide range of real-world problems, such as data analysis on sensor networks, biological networks, and neural networks. In this project, the investigator uses a blend of mathematical theories and techniques to develop the theoretical underpinning and possible new applications of GSP, especially for the case of large dynamic networks that evolve over time. The investigator plans to couple this research with graduate student mentoring, organizing scientific workshops, and outreach in the scientific community. In this project, the investigator aims to leverage a blend of techniques from harmonic analysis, functional analysis, and graph-limit theory to address challenges in information processing, particularly in the theoretical underpinning of GSP. The goal is to develop a theory that is applicable to a wide range of large dynamic networks. The relatively recent theory of graph limits and graphons provides a valuable non-parametric approach to modeling networks, particularly for stochastic networks. Indeed, graphons represent random processes that generate networks, and networks produced by the same graphon have similar large-scale features. Many large networks that arise naturally are manifestations of an underlying (hidden) spatial reality and can be efficiently modeled using Cayley graphons. In this project, the investigator develops signal processing on Cayley graphons as well as instance-independent GSP for samples of Cayley graphons. Current instance-independent graphon-based signal-processing methods apply to only dense undirected graphs. To develop a theory that applies to a large class of (possibly sparse or directed) networks, the investigator plans to address several theoretical challenges, such as convergence of spectra for sparse/directed graphs and spectral theory of nearly Cayley graphs. 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.