Contributions to Persistence Theory

Persistence theory discussed in this paper is an application of algebraic topology (Morse Theory [29]) to Data Analysis, precisely to qualitative understanding of point cloud data, or PCD for short. PCD can be geometrized as a filtration of simplicial complexes (Vietoris-Rips complex [25] [36]) and the homology changes of these complexes provide qualitative information about the data. Bar codes describe the changes in homology with coefficients in a fixed field. 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 [20]). When the coefficient field is ℝ, we propose an algorithm based on the Hodge decomposition [17]. With Dan Burghelea and Tamal K. Dey we developed a persistence theory which involves level sets discussed in Section 4. 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 bar codes for level persistence. We provide enhancements and modifications of ELZ algorithm to calculate such invariants and illustrate them by examples.