Contributions to Persistence Theory

Persistence theory discussed in this paper 1 is an application of algebraic topology (Morse Theory) to Data Analysis, precisely to qualitative understanding of point cloud data. Mathematically a point cloud data is a finite metric space of a very large cardinality. It can be geometrized as a filtration of simplicial complexes and the homology changes of these complexes provide qualitative information about the data. There are new invariants which permit to describe the changes in homology (with coefficients in a fixed field) and these invariants are the “bar codes”. In Section 3 we develop additional methods for the calculation of bar codes and their refinements. When the coefficient field is Z2, the calculation of bar codes is done by ELZ algorithm (named after H. Edelsbrunner, D. Letscher, and A. Zomorodian). When the coefficient field is R, we propose an algorithm based on the Hodge decomposition. The original persistence theory involves the “sub-level sets” of a nice continuous map (tame map). With Dan Burghelea and Tamal Dey we developed a persistence theory which involves level sets discussed in Section 4. This is a refinement of the original persistence. The level persistence we propose is an alternative to Zigzag persistence considered by G. Carlsson and V. D. Silva. We introduce and discuss new computable invariants, the “relevant level persistence numbers” and the “positive and negative bar codes”, and explain how they are related to the known ones. We provide enhancements and modifications of ELZ algorithm to calculate such invariants and illustrate them by examples. Sections 3 is preceded by background materials (Section 2) where the concepts of algebraic topology used in this paper are defined.