# Geometry in the space of persistence modules

@inproceedings{Silva2013GeometryIT, title={Geometry in the space of persistence modules}, author={Vin de Silva and Vidit Nanda}, booktitle={SoCG '13}, year={2013} }

Topological persistence is, by now, an established paradigm for constructing robust topological invariants from point cloud data: the data are converted into a filtered simplicial complex, the complex gives rise to a persistence module, and the module is described by a persistence diagram. In this paper, we study the geometry of the spaces of persistence modules and diagrams, with special attention to Cech and Rips complexes. The metric structures are determined in terms of interleaving maps…

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## References

SHOWING 1-10 OF 31 REFERENCES

Proximity of persistence modules and their diagrams

- Mathematics, Computer ScienceSCG '09
- 2009

This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.

Stability of persistence diagrams

- MathematicsSCG
- 2005

The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram…

Categorification of Persistent Homology

- MathematicsDiscret. Comput. Geom.
- 2014

This work redevelops persistent homology (topological persistence) from a categorical point of view and gives a natural construction of a category of ε-interleavings of $\mathbf {(\mathbb {R},\leq)}$-indexed diagrams in some target category and shows that if the target category is abelian, so is this category of interleavments.

Topological persistence and simplification

- EconomicsProceedings 41st Annual Symposium on Foundations of Computer Science
- 2000

A notion of topological simplification is formalized within the framework of a filtration, which is the history of a growing complex, and a topological change that happens during growth is classified as either a feature or noise, depending on its life-time or persistence within the filTration.

Vines and vineyards by updating persistence in linear time

- Mathematics, Computer ScienceSCG '06
- 2006

The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering and uses the algorithm to compute 1-parameter families of diagrams which are applied to the study of protein folding trajectories.

Computing persistent homology

- Mathematics, Computer ScienceSCG '04
- 2004

The homology of a filtered d-dimensional simplicial complex K is studied as a single algebraic entity and a correspondence is established that provides a simple description over fields that enables a natural algorithm for computing persistent homology over an arbitrary field in any dimension.

Persistent Homology — a Survey

- Mathematics

Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical…

Fast construction of the Vietoris-Rips complex

- Computer Science, MathematicsComput. Graph.
- 2010

Probability measures on the space of persistence diagrams

- Mathematics
- 2011

This paper shows that the space of persistence diagrams has properties that allow for the definition of probability measures which support expectations, variances, percentiles and conditional…