SPECTRAL SPARSIFICATION FROM FIRST PRINCIPLES
Date
2018-05
Authors
Journal Title
Journal ISSN
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Publisher
University of Delaware
Abstract
Graphs are abstract mathematical objects used to model networks, and spectral
graph theory is the sub eld studying matrices, eigenvalues, and eigenvectors associated
with graphs. We consider the spectral graph sparsi cation: the problem of construct-
ing sparse approximations of dense graphs with respect to spectral characteristics. We
provide an exposition of this problem from the ground up. We explore two motivations
for this problem: as a means to contains data explosion and as a generalization of
expander graphs. From here, we formalize the problem and provide three probabilistic
algorithms to construct spectral sparsi ers. This is done with incremental improve-
ment to demonstrate the intuitions and proof techniques underlying sparsi cation al-
gorithms. The rst algorithm does simple random sampling with xed probability, and
the other two algorithms make use of e ective resistances, of which the latter employs
a law of large numbers technique.
Description
Keywords
computer science, spectal sparsification, first principles