The Structure and Stability of Persistence Modules

Frédéric Chazal, Vin de Silva, Marc Glisse, Steve Oudot (Autoren)

Buch | Softcover

X, 120 Seiten

2016 | 1st ed. 2016
Springer International Publishing (Verlag)
978-3-319-42543-6 (ISBN)

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This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such modules, providing a sound mathematical framework for the study of persistence diagrams. Completely self-contained, this brief introduces the notion of persistence measure and makes extensive use of a new calculus of quiver representations to facilitate explicit computations.
Appealing to both beginners and experts in the subject, The Structure and Stability of Persistence Modules provides a purely algebraic presentation of persistence, and thus complements the existing literature, which focuses mainly on topological and algorithmic aspects.

Introduction.- Persistence Modules.- Rectangle Measures.- Interleaving.- The Isometry Theorem.- Variations.- References.- Index.

"This book is a very nice contribution to the subject of Topological Data Analysis. In this slim volume, the novice will find a collection of main results with their proofs and many references; additionally, experts will see persistence developed more generally than usual using measure theory. ... There are many synthesizing comments throughout the text to help the reader put the material in context, and the writing itself is lucid." (Michele Intermont, MAA Reviews, October, 2017)

"This monograph develops the theory of persistence modules over the real line in a manner that is well-motivated, accessible, thorough, and self-contained." (Henry Hugh Adams, Mathematical Reviews, October, 2017)

"This book offers an excellent introduction to anyone interested in understanding the fundamentals of persistent homology. The exposition is clear, concise and easy to read. ... A fair overview of similar results appearing elsewhere is given, and an extensive list of suggested further reading is provided for the inspired reader. ... In short, the book offers a self-contained introduction to topics such as persistence modules, persistence diagrams, interleavings, and the famous algebraic stability theorem." (Magnus Bakke Botnan, zbMATH, 2017)

"This monograph develops the theory of persistence modules over the real line in a manner that is well-motivated, accessible, thorough, and self-contained. ... In this monograph, the theory of persistence modules over the reals is presented from scratch, with the main results and their proofs in a natural framework that is convenient to learn and to use." (Henry Hugh Aams, Mathematical Reviews, 2017)

Erscheinungsdatum	19.08.2016
Reihe/Serie	SpringerBriefs in Mathematics
Zusatzinfo	X, 120 p. 17 illus., 15 illus. in color.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Informatik ► Theorie / Studium
	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Algebra • algebraic topology • applied algebraic topology • Mathematical Applications in Computer Science • mathematics and statistics • persistence diagrams • persistence modules • persistence stability theorem • quiver representations • topological data analysis • topological persistence
ISBN-10	3-319-42543-9 / 3319425439
ISBN-13	978-3-319-42543-6 / 9783319425436
Zustand	Neuware