Zigzag Persistent Homology Survey Ian Frankenburg

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

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

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.

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

Category Theory A functor is a map between categories that preserves structure. For categories C and D, functor satifies the following: 1. 2. 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

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

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

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

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

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