Localized Spectral Graph Filter Frames: A Unifying Framework, Survey of Design Considerations, and Numerical Comparison
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by
David I Shuman
2020
Abstract
Representing data residing on a graph as a linear combination of building
block signals can enable efficient and insightful visual or statistical
analysis of the data, and such representations prove useful as regularizers in
signal processing and machine learning tasks. Designing such collections of
building block signals -- or more formally, dictionaries of atoms -- that
specifically account for the underlying graph structure as well as any
available representative training signals has been an active area of research
over the last decade. In this article, we survey a particular class of
dictionaries called localized spectral graph filter frames, whose atoms are
created by localizing spectral patterns to different regions of the graph.
After showing how this class encompasses a variety of approaches from spectral
graph wavelets to graph filter banks, we focus on the two main questions of how
to design the spectral filters and how to select the center vertices to which
the patterns are localized. Throughout, we emphasize computationally efficient
methods that ensure the resulting transforms and their inverses can be applied
to data residing on large, sparse graphs. We demonstrate how this class of
transform methods can be used in signal processing tasks such as denoising and
non-linear approximation, and provide code for readers to experiment with these
methods in new application domains.
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