Zigzag persistent homology and real-valued functions

We study the problem of computing zigzag persistence of a sequence of homology groups and study a particular sequence derived from the levelsets of a real-valued function on a topological space. The result is a local, symmetric interval descriptor of the function. Our structural results establish a connection between the zigzag pairs in this sequence and extended persistence, and in the process resolve an open question associated with the latter. Our algorithmic results not only provide a way to compute zigzag persistence for any sequence of homology groups, but combined with our structural results give a novel algorithm for computing extended persistence. This algorithm is easily parallelizable and uses (asymptotically) less memory. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems; G.2.1 [Discrete Mathematics]: Combinatorics—Counting problems General Terms algorithms, theory Keywords Zigzag persistent hom...

• Type: Conference
• Year: 2009
• Where: COMPGEOM
• Authors: Gunnar Carlsson, Vin de Silva, Dmitriy Morozov
• Keywords: COMPGEOM 2009, Discrete Geometry, Extended Persistence, Homology Groups, Zigzag Persistence