# Persistent Homology — a Survey

@inproceedings{EdelsbrunnerPersistentH, title={Persistent Homology — a Survey}, author={Herbert Edelsbrunner and John Harer} }

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 formalism. Here we give a record of the short history of persistent homology and present its basic concepts. Besides the mathematics we focus on algorithms and mention the various connections to applications, including to biomolecules, biological networks, data analysis, and geometric modeling.

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## 823 Citations

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It turns out that these a priori known facts are clearly retrieved by persistent homology analysis of dynamically sampled submanifolds of configuration space.

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## References

SHOWING 1-10 OF 54 REFERENCES

Topological Persistence and Simplification

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

Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.

Computing Persistent Homology

- Mathematics, Computer ScienceSCG '04
- 2004

The analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields and derives an algorithm for computing individual persistent homological groups over an arbitrary principal ideal domain in any dimension.

The Theory of Multidimensional Persistence

- MathematicsSCG '07
- 2007

This paper proposes the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and proves its completeness in one dimension.

Localized Homology

- Computer ScienceIEEE International Conference on Shape Modeling and Applications 2007 (SMI '07)
- 2007

Persistence barcodes for shapes

- Mathematics, Computer ScienceSGP '04
- 2004

This paper initiates a study of shape description and classification via the application of persistent homology to two tangential constructions on geometric objects, obtaining a shape descriptor, called a barcode, that is a finite union of intervals.

Stability of Persistence Diagrams

- MathematicsDiscret. Comput. Geom.
- 2007

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…

Extending Persistence Using Poincaré and Lefschetz Duality

- MathematicsFound. Comput. Math.
- 2009

An algebraic formulation is given that extends persistence to essential homology for any filtered space, an algorithm is presented to calculate it, and how it aids the ability to recognize shape features for codimension 1 submanifolds of Euclidean space is described.

Computing Linking Numbers of a Filtration

- Computer ScienceWABI
- 2001

We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of non-trivial tangling…

Weak feature size and persistent homology: computing homology of solids in Rn from noisy data samples

- MathematicsSCG
- 2005

It is proved that under quite general assumptions one can deduce the topology of a bounded open set in Rn from a Hausdorff distance approximation of it and the weak feature size (wfs) is introduced that generalizes the notion of local feature size.

Size Functions from a Categorical Viewpoint

- Mathematics
- 2001

A new categorical approach to size functions is given. Using this point of view, it is shown that size functions of a Morse map, f: M→ℜ can be computed through the 0-dimensional homology. This result…