# The Theory of the Interleaving Distance on Multidimensional Persistence Modules

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## Abstract

In 2009, Chazal et al. introduced \(\epsilon \)-*interleavings* of persistence modules. \(\epsilon \)-interleavings induce a pseudometric \(d_\mathrm{I}\) on (isomorphism classes of) persistence modules, the *interleaving distance*. The definitions of \(\epsilon \)-interleavings and \(d_\mathrm{I}\) generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view toward applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, \(d_\mathrm{I}\) is equal to the bottleneck distance \(d_\mathrm{B}\). This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the *isometry theorem*. Second, we present a characterization of the \(\epsilon \)-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two \(\epsilon \)-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, \(d_\mathrm{I}\) satisfies a universality property. This universality result is the central result of the paper. It says that \(d_\mathrm{I}\) satisfies a stability property generalizing one which \(d_\mathrm{B}\) is known to satisfy, and that in addition, if \(d\) is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then \(d\le d_\mathrm{I}\). We also show that a variant of this universality result holds for \(d_\mathrm{B}\), over arbitrary fields. Finally, we show that \(d_\mathrm{I}\) restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.

## Keywords

Multidimensional persistence Stability of persistent homology Persistence modules Interleavings Algebraic stability Isometry theorem## Mathematics Subject Classification

55 68## Notes

### Acknowledgments

The first version of this paper was written while I was a graduate student. Discussions with my Ph.D. adviser Gunnar Carlsson catalyzed the research presented here in several ways. In addition, Gunnar served as a patient and helpful sounding board for the ideas of this paper. I thank him for his support and guidance. Thanks to Henry Adams, Peter Bubenik, Patrizio Frosini, Peter Landweber, Dmitriy Morozov, and the anonymous referees for useful corrections and helpful feedback on this work. Parts of the exposition in Sects. 1.1 and 3 benefited from edits done jointly with Ulrich Bauer on closely related material in [3]. The main result of William Crawley-Boevey’s paper [26] plays an important role in the present version of this work. I thank Bill for writing his paper and both Bill and Vin de Silva for enlightening discussions about structure theorems for \({\mathbb {R}}\)-graded persistence modules. Thanks to Stanford University, the Technion, the Institute for Advanced Study, and the Institute for Mathematics and its Applications for their support hospitality during the writing and revision of this paper. This work was supported by ONR grant N00014-09-1-0783 and NSF grant DMS-1128155. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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