The Structure and Stability of Persistence Modules
By Frédéric Chazal, Vin de Silva, Marc Glisse, Steve Y. Oudot. Springer Briefs in Mathematics, 2012
528 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/8fade1333b64ab0e14d0ff8b0fbf8b82fbdb434d
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In Algebra and Number Theory
- Frédéric Chazal
- Vin de Silva
- W. Crawley‐Boevey
- Ulrich Bauer
- Vladimir Petrov Kostov
- Daisuke Yamakawa
- V. Ginzburg
- I. Losev
In Computational Theory and Mathematics
- Frédéric Chazal
- Larry A. Wasserman
- Ulrich Bauer
- Bertrand Michel
- Peter Bubenik
- Vin de Silva
- W. Crawley‐Boevey
- Yen-Chi Chen
In Mathematical Physics
- Frédéric Chazal
- Vin de Silva
- Peter Bubenik
- W. Crawley‐Boevey
- Larry A. Wasserman
- Ulrich Bauer
- Elizabeth Munch
- Bertrand Michel
Citations in the corpus
- Asymptotic theory for density ridges (2014, background): “Second, as is shown in Genovese et al. (2014), when a topological assumption called tameness is assumed [CohenSteiner, Edelsbrunner and Harer (2007), Chazal et al. (2009, 2012)], then Rh and R have the same topology for small h.”
- Theory of interleavings on categories with a flow (2017, background): “In [3], the stability of parts (i) and (ii) are respectively called ‘soft’ stability and ‘hard’ stability: part (i) operates at an abstract algebraic/categorical level, while part (ii) requires a detailed study of the relationship between persistence modules and their diagrams [7, 5, 6, 1].”
- Statistical topological data analysis using persistence landscapes (2012, background): “All of the information in a persistence module is completely contained in a multiset of intervals called a barcode [47, 23, 15].”
- Ripser: efficient computation of Vietoris–Rips persistence barcodes (2019, background): “Applying homology to a filtration of finite simplicial complexes • : → Simp yields another diagram ∗( •) : → Vect? of finite dimensional vector spaces over F?, often called a persistence module [7].”
- Wasserstein Stability for Persistence Diagrams (2020, unknown): “While a complete overview of these results is beyond the scope of this paper, key references include the original stabilty result [16], and its algebraic and categorical generalizations in [10] and [8] respectively.”
- The observable structure of persistence modules (2014, methodology): “Associated to a persistence module V is its persistence measure [2].”
- An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, background): “When a persistence module V can be decomposed as a direct sum of interval modules, one can show that this decomposition is unique up to reordering the intervals (see (Chazal et al., 2016a, Theorem 2.7)).”