  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 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.
