Browsing by Author "Roddenberry, T. Mitchell"

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    Sampling and Limit Theories for Graph Signal Processing and Large Simplicial Complexes
    (2023-03-31) Roddenberry, T. Mitchell; Segarra, Santiago
    This thesis considers the role of locality and sampling in graph signal processing and network science. In light of the prevalence of extremely large and complex network datasets, it is a timely problem to consider how the study of these objects can be reduced to the study of a distribution of simpler objects. Indeed, many methods in graph signal processing and machine learning on graphs can be described strictly in light of local graph substructures. The approach taken to this problem starts with defining the notion of taking a sample from a network, and then builds this out to a probabilistic framework for network theory. This framework is applied to understand questions of graph parameter estimation, fundamentals of graph signal processing and Fourier analysis on graphs, transferability of machine learning and signal processing methods for network data, and finally to construct meaningful limiting objects of graphs and simplicial complexes. The study undertaken in this thesis both enhances the understanding of current methods, as well as inspires new methods in light of the notions of transferability defined by sampling.
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    Signal processing on higher-order networks: Livin’ on the edge... and beyond
    (Elsevier, 2021) Schaub, Michael T.; Zhu, Yu; Seby, Jean-Baptiste; Roddenberry, T. Mitchell; Segarra, Santiago
    In this tutorial, we provide a didactic treatment of the emerging topic of signal processing on higher-order networks. Drawing analogies from discrete and graph signal processing, we introduce the building blocks for processing data on simplicial complexes and hypergraphs, two common higher-order network abstractions that can incorporate polyadic relationships. We provide brief introductions to simplicial complexes and hypergraphs, with a special emphasis on the concepts needed for the processing of signals supported on these structures. Specifically, we discuss Fourier analysis, signal denoising, signal interpolation, node embeddings, and nonlinear processing through neural networks, using these two higher-order network models. In the context of simplicial complexes, we specifically focus on signal processing using the Hodge Laplacian matrix, a multi-relational operator that leverages the special structure of simplicial complexes and generalizes desirable properties of the Laplacian matrix in graph signal processing. For hypergraphs, we present both matrix and tensor representations, and discuss the trade-offs in adopting one or the other. We also highlight limitations and potential research avenues, both to inform practitioners and to motivate the contribution of new researchers to the area.