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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.

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