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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825 Citations
Highly Influential Citations
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379
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210
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825 Citations

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TLDR
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The use of persistent homology methods to capture structural and topological properties of graphs and use it to address the problem of link prediction is proposed.
Persistent homology of directed networks
TLDR
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Persistent homology analysis of phase transitions.
TLDR
It turns out that these a priori known facts are clearly retrieved by persistent homology analysis of dynamically sampled submanifolds of configuration space.
Persistent homology for random fields and complexes
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When shapes of objects are modeled as topological spaces endowed with functions, the shape comparison problem can be dealt with using persistent homology to provide shape descriptors, and the…
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References

SHOWING 1-10 OF 56 REFERENCES
Computing Persistent Homology
TLDR
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.
The Theory of Multidimensional Persistence
TLDR
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
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    IEEE International Conference on Shape Modeling and Applications 2007 (SMI '07)
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Persistence barcodes for shapes
TLDR
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
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…
Weak feature size and persistent homology: computing homology of solids in Rn from noisy data samples
TLDR
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
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…
Vines and vineyards by updating persistence in linear time
TLDR
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.
Intersection Homology
Extreme Elevation on a 2-Manifold
TLDR
An algorithm for finding points of locally maximum elevation, which is invariant under rigid motions and can be used to define features such as lines of discontinuous or continuous but non-smooth elevation.
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