1. Multidimensional Persistence and Noise
- Author
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Martina Scolamiero, Sebastian Öberg, Ryan Ramanujam, Wojciech Chachólski, and Anders Lundman
- Subjects
Computational Geometry (cs.CG) ,FOS: Computer and information sciences ,Multidimensional persistence ,Pure mathematics ,Noise systems ,010103 numerical & computational mathematics ,02 engineering and technology ,Pseudometric space ,Stable invariants ,Commutative Algebra (math.AC) ,Barcode ,01 natural sciences ,law.invention ,law ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Persistence modules ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Functor ,Applied Mathematics ,Numerical analysis ,Mathematics - Commutative Algebra ,Computational Mathematics ,Computational Theory and Mathematics ,Computer Science - Computational Geometry ,020201 artificial intelligence & image processing ,Analysis - Abstract
In this paper we study multidimensional persistence modules [5,13] via what we call tame functors and noise systems. A noise system leads to a pseudo-metric topology on the category of tame functors. We show how this pseudo-metric can be used to identify persistent features of compact multidimensional persistence modules. To count such features we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For 1-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction., Comment: Found Comput Math (2016)
- Published
- 2016