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
Highly Influential Citations
60
Background Citations
382
Methods Citations
208
Results Citations
3

823 Citations

Introduction to Persistent Homology
This video presents an introduction to persistent homology, aimed at a viewer who has mathematical aptitude but not necessarily knowledge of algebraic topology. Persistent homology is an algebraic…
A Kernel for Multi-Parameter Persistent Homology
A Persistent Homology Perspective to the Link Prediction Problem
TLDR
This work proposes the use of persistent homology methods to capture structural and topological properties of graphs and use it to address the problem of link prediction.
Confidence sets for persistence diagrams
TLDR
This paper derives confidence sets that allow us to separate topological signal from topological noise, and brings some statistical ideas to persistent homology.
Understanding and Predicting Links in Graphs: A Persistent Homology Perspective
TLDR
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
A method for constructing simplicial complexes from weighted, directed networks that captures directionality information is studied, and it is able to prove that the persistent homology of such complexes is stable with respect to a certain notion of network distance.
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.
Stable shape comparison by persistent homology
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…
One-dimensional reduction of multidimensional persistent homology
A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a…
Persistent homology and applied homotopy theory
This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence…
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References

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TLDR
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.
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TLDR
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
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.
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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…
Extending Persistence Using Poincaré and Lefschetz Duality
TLDR
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
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
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…
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