Topological data analysis (TDA) has emerged recently as a viable
tool for analyzing complex data, and the area has grown
substantially both in its methodologies and applicability.
Providing a computational and algorithmic foundation for techniques
in TDA, this comprehensive, self-contained text introduces students
and researchers in mathematics and computer science to the current
state of the field. The book features a description of mathematical
objects and constructs behind recent advances, the algorithms
involved, computational considerations, as well as examples of
topological structures or ideas that can be used in applications.
It provides a thorough treatment of persistent homology together
with various extensions - like zigzag persistence and
multiparameter persistence - and their applications to different
types of data, like point clouds, triangulations, or graph data.
Other important topics covered include discrete Morse theory, the
Mapper structure, optimal generating cycles, as well as recent
advances in embedding TDA within machine learning frameworks.
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