Topological Data Analysis - Multiscale Invariants

Multiscale Invariants

1. Multifiltration Model. Morse Theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
2. Persistent homology

See homology for an introduction to the notation. Persistent homology essentially calculates homology groups at different spatial resolutions to see which features persist over a wide range of length scales. It is assumed that important features and structures are the ones that persist. We define persistent homology as follows: Let be a filtration. The p-persistent kth homology group of is .

If we let be a nonbounding -cycle created at time by simplex and let be a homologous -cycle that becomes a boundary cycle at time by simplex, then we can define the persistence interval associated to as . We call the creator of and the destroyer of . If does not have a destroyer, its persistence is . Instead of using an index-based filtration, we can use a time-based filtration. Let be a simplicial complex and be a filtration defined for an associated map that maps simplices in the final complex to real numbers. Then for all real numbers, the -persistent kth homology group of is . The persistence of a -cycle created at time and destroyed at is . There are various software packages for computing persistence intervals of a finite filtration, such as jPlex, Dionysus, Perseus, and PHAT.