Zigzag Persistent Homology Survey Ian Frankenburg Review Of

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 Image Credit: Robert Ghrist • Persistent Homology is computable via linear algebra

Review Continued 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