Persistent Homology — a Survey

  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. 
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 Persistent Homology Perspective to the Link Prediction Problem
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
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
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
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.
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
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more
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


Computing Persistent Homology
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
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
  • A. Zomorodian, G. Carlsson
  • Computer Science
    IEEE International Conference on Shape Modeling and Applications 2007 (SMI '07)
  • 2007
Persistence barcodes for shapes
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
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
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
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.