Computing Optimal Persistent Cycles for Levelset Zigzag on Manifold-like Complexes

05/02/2021
by   Tamal K. Dey, et al.
0

In standard persistent homology, a persistent cycle born and dying with a persistence interval (bar) associates the bar with a concrete topological representative, which provides means to effectively navigate back from the barcode to the topological space. Among the possibly many, optimal persistent cycles bring forth further information due to having guaranteed quality. However, topological features usually go through variations in the lifecycle of a bar which a single persistent cycle may not capture. Hence, for persistent homology induced from PL functions, we propose levelset persistent cycles consisting of a sequence of cycles that depict the evolution of homological features from birth to death. Our definition is based on levelset zigzag persistence which involves four types of persistence intervals as opposed to the two types in standard persistence. For each of the four types, we present a polynomial-time algorithm computing an optimal sequence of levelset persistent p-cycles for the so-called weak (p+1)-pseudomanifolds. Given that optimal cycle problems for homology are NP-hard in general, our results are useful in practice because weak pseudomanifolds do appear in applications. Our algorithms draw upon an idea of relating optimal cycles to min-cuts in a graph that we exploited earlier for standard persistent cycles. Note that levelset zigzag poses non-trivial challenges for the approach because a sequence of optimal cycles instead of a single one needs to be computed in this case.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
10/11/2018

Persistent 1-Cycles: Definition, Computation, and Its Application

Persistence diagrams, which summarize the birth and death of homological...
research
01/03/2021

Cycle Registration in Persistent Homology with Applications in Topological Bootstrap

In this article we propose a novel approach for comparing the persistent...
research
12/14/2017

Volume Optimal Cycle: Tightest representative cycle of a generator on persistent homology

This paper shows a mathematical formalization, algorithms and computatio...
research
12/04/2021

On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class

Homology features of spaces which appear in applications, for instance 3...
research
12/05/2022

Connecting Discrete Morse Theory and Persistence: Wrap Complexes and Lexicographic Optimal Cycles

We study the connection between discrete Morse theory and persistent hom...
research
08/10/2022

Stable Homology-Based Cycle Centrality Measures for Weighted Graphs

Network centrality measures are concerned with evaluating the importance...
research
07/10/2019

Computing Minimal Persistent Cycles: Polynomial and Hard Cases

Persistent cycles, especially the minimal ones, are useful geometric fea...

Please sign up or login with your details

Forgot password? Click here to reset