# The Theory of the Interleaving Distance on Multidimensional Persistence Modules

@article{Lesnick2015TheTO, title={The Theory of the Interleaving Distance on Multidimensional Persistence Modules}, author={Michael Lesnick}, journal={Foundations of Computational Mathematics}, year={2015}, volume={15}, pages={613-650} }

In 2009, Chazal et al. introduced $$\epsilon $$ϵ-interleavings of persistence modules. $$\epsilon $$ϵ-interleavings induce a pseudometric $$d_\mathrm{I}$$dI on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $$\epsilon $$ϵ-interleavings and $$d_\mathrm{I}$$dI 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…

## 174 Citations

Tracking the variety of interleavings.

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In topological data analysis persistence modules are used to distinguish the legitimate topological features of a finite data set from noise. Interleavings between persistence modules feature…

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A polynomial time algorithm is presented to compute the bottleneck distance for modules from indecomposables, which bounds the interleaving distance from above, and another algorithm is given to compute a new distance called {\em dimension distance} that bounds it from below.

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Decomposition of Exact pfd Persistence Bimodules

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We characterize the class of persistence modules indexed over $$\mathbb {R}^2$$ R 2 that are decomposable into summands whose supports have the shape of a block —i.e. a horizontal band, a vertical…

Computing Bottleneck Distance for 2-D Interval Decomposable Modules

- Computer Science, MathematicsSoCG
- 2018

A polynomial time algorithm is presented to compute the bottleneck distance for modules from indecomposables, which bounds the interleaving distance from above, and another algorithm is given to compute a new distance called {\em dimension distance} that bounds it from below.

Universality of the Homotopy Interleaving Distance

- MathematicsArXiv
- 2017

It is observed that any pseudometric satisfying natural stability and homotopy invariance axioms can be used to formulate lifts of several fundamental TDA theorems from the algebraic (homological) level to the level of filtered spaces.

Induced matchings and the algebraic stability of persistence barcodes

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- 2015

This work shows explicitly how a $\delta$-interleaving morphism between two persistence modules induces a $delta-matching between the barcodes of the two modules, and yields a novel "single-morphism" characterization of the interleaving relation on persistence modules.

Generalized persistence diagrams

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- 2018

The persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer is generalized to the setting of constructible persistence modules valued in a symmetric monoidal category and a second type of persistence diagram is defined, which enjoys a stronger stability theorem.

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It is shown that on 1or 2-parameter persistence modules over prime fields, dp I is the universal metric satisfying a natural stability property; this result extends a stability result of Skraba and Turner for the p-Wasserstein distance on barcodes in the 1- parameter case, and is also a close analogue of a universality property for the interleaving distance given by the second author.

Theory of interleavings on categories with a flow

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- 2017

The interleaving distance was originally defined in the field of Topological Data Analysis (TDA) by Chazal et al. as a metric on the class of persistence modules parametrized over the real line.…

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