Paper Info
Title
Volume-Optimal Cycle: Tightest Representative Cycle Of A Generator In Persistent Homology
Abstract
The present paper shows a mathematical formalization of-as well as algorithms and software for computing-volume-optimal cycles. Volume-optimal cycles are useful for understanding geometric features appearing in a persistence diagram. Volume-optimal cycles provide concrete and optimal homologous structures, such as rings or cavities, on a given dataset. The key idea is the optimality on a (q + 1)-chain complex for a qth homology generator. This optimality formalization is suitable for persistent homology. We can solve the optimization problem using linear programming. For an alpha filtration on R-n, volume-optimal cycles on an (n - 1) st persistence diagram are more efficiently computable using a merge-tree algorithm. The merge-tree algorithm also provides a tree structure on the diagram containing richer information than volume-optimal cycles. The key mathematical idea used here is Alexander duality.
Year
DOI
Venue
2018
10.1137/17M1159439
SIAM JOURNAL ON APPLIED ALGEBRA AND GEOMETRY
Keywords
Field
DocType
persistent homology, algebraic topology, optimization
Topology,Discrete mathematics,Persistent homology,Diagram,Software,Linear programming,Tree structure,Optimization problem,Mathematics,Alexander duality,Computation
Journal
Volume
Issue
ISSN
2
4
2470-6566
Citations 
PageRank 
References 
1
0.36
2
Authors
1
Name
Order
Citations
PageRank
ippei obayashi151.92