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The theory of the interleaving distance on multidimensional persistence modules. (English) Zbl 1335.55006
Roughly speaking, one-dimensional persistent topology is about filtered spaces, indexed by real numbers [H. Edelsbrunner and J. Harer, Contemp. Math. 453, 257–282 (2008; Zbl 1145.55007)]. One of the main tools is the persistence module, a diagram of vector spaces also indexed by real numbers, usually attached to a filtration through homology [A. Zomorodian and G. Carlsson, Discrete Comput. Geom. 33, No. 2, 249–274 (2005; Zbl 1069.55003)]. A persistence module (PM) \(M\), in turn, can be dealt with through a barcode \(\mathcal B_M\), a collection of intervals which characterizes \(M\) if its vector spaces are finite dimensional [G. Carlsson et al., Int. J. Shape Model. 11, No. 2, 149–187 (2005; Zbl 1092.68688)].
Essentially, two PMs \(M\) and \(N\) are \(\varepsilon\)-interleaved if they correspond up to \(\varepsilon\)-shifts; if \(0 \leq \varepsilon \leq \varepsilon'\) and \(M, N\) are \(\varepsilon\)-interleaved, then they are also \(\varepsilon'\)-interleaved. The interleaving distance \(d_I\) between \(M\) and \(N\) is the infimum of such \(\varepsilon\); \(d_I\) is an extended pseudometric [F. Chazal et al., “Proximity of persistence modules and their diagrams”, in: Proceedings of the 25th annual symposium on computational geometry, SCG 2009, Aarhus, Denmark, June 8–10, 2009. New York, NY: Association for Computing Machinery (ACM). xii, 413 p. (2009; Zbl 1271.68008)].
The present paper extends the theory of \(d_I\) to the case of multidimensional indexing of the PMs and of the filtrations [G. Carlsson and A. Zomorodian, Discrete Comput. Geom. 42, No. 1, 71–93 (2009; Zbl 1187.55004)], where the PMs have a more tricky structure. The first relevant result is, however, in the 1D case: it is a new proof of the isometry theorem, the equality between \(d_I(M, N)\) and the bottleneck distance of the corresponding barcodes.
The second important result is a characterization of \(\varepsilon\)-interleaving of PMs with nD indexing: it reflects upon \(\varepsilon\)-shifts of generators and relators of the modules.
The universality result is a corollary of the characterization; it says that for multidimensional PMs over a prime field \(d_I\) is stable and optimal; it generalizes a result of [M. d’Amico et al., Acta Appl. Math. 109, No. 2, 527–554 (2010; Zbl 1198.68224)] and makes use of the natural pseudodistance between spaces endowed with filtering functions, defined in the nD case in [P. Frosini and M. Mulazzani, Bull. Belg. Math. Soc. - Simon Stevin 6, No. 3, 455–464 (1999; Zbl 0937.55010)]. It has a beautiful topological proof and is conjectured to hold for any field.
The fourth main result of the paper asserts that, also in the nD case, the set of real numbers \(\varepsilon\) for which two finitely presented PMs are \(\varepsilon\)-interleaved is closed.
Priorities and contributions are duly acknowledged. Well-chosen, simple examples clarify the concepts. The extensive use of the categorical setting requires an attentive reading but grants indisputable and general results.

55N99 Homology and cohomology theories in algebraic topology
54E35 Metric spaces, metrizability
68R99 Discrete mathematics in relation to computer science
Full Text: DOI arXiv
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