1. Zigzag Persistent Homology Survey
Ian Frankenburg

2. Review Of Standard Persistence
Algebraic method for discerning topological features of data (holes, clusters, etc)
Given a point cloud of data, construct a simplicial complex
Image Credit: Robert Ghrist

3. Review Continued
Different complexes appear at different ε values, so track how complex becomes connected with barcodes
Persistent Homology is computable via linear algebra
Image Credit: Robert Ghrist

4. Review Continued Each simplicial complex is a subcomplex of the next
Image Credit: Robert Ghrist
Each simplicial complex is a subcomplex of the next
Sequence of simplicial complexes is called a filtration
Applying homology to a filtration results in an algebraic structure known as persistence module.

5. Category Theory
A category C consists of: 1. a class of objects, denoted Obj(C) 2. a class of maps between objects, called morphisms
The Morphisms between objects must be associative and there must exists an identity morphism for each object
Examples: 1. the category Sets with morphisms being functions from one set to another 2. the category Top of topological spaces with morphisms being continuous maps between spaces

6. Category Theory
A functor is a map between categories that preserves structure. For categories C and D, functor satifies the following: F preserves composition and identity morphisms
Examples: 1. A forgetful functor neglects some or all of the input properties. Mapping the category Group to Sets where the mathematical group is mapped to its underlying set is a forgetful functor 2. Homology is a functor from topological spaces to the category of chain complexes

7. Zigzag Persistence
Zigzag Persistence is the algebraic generalization of standard persistent homology. Below is a zigzag diagram between vector spaces connected with linear maps
Applying the homology functor then gives

8. Zigzag Persistence
The zigzag modules can then be decomposed uniquely into interval modules, so the total persistence of the zigzag diagram is a collection of multisets
This is important for theoretical foundations as well as algorithmic implementation

9. Topological Bootstrapping
Statistical Bootstrapping is any test that relies on random sampling with replacement
Analogously, topological bootstrapping involves estimating topological structure based on samples
Given samples

10. Topological Bootstrapping
Applying the homology functor gives
Zigzag persistence then provides a way to understand which features are measured and persist by different samples
Algorithm for topological bootstrapping relies on using add and remove subroutines. Technical but accessible in Dr. Carlsson’s paper

11. Witness Complexes
Čech and Vietoris-Rips complexes can often be massive
Can define a biwitness complex which maps to two witness complexes
Given a set of landmark points, can then construct the zigzag complex
Long intervals in the zigzag barcode will indicate features that persist across choices of landmarks