Persistent homology captures the topology of a filtration-a one-parameter family of increasing spaces-in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence. Instead, we propose the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and prove its completeness in one dimension. © 2009 Springer Science+Business Media, LLC.
CITATION STYLE
Carlsson, G., & Zomorodian, A. (2009). The theory of multidimensional persistence. Discrete and Computational Geometry, 42(1), 71–93. https://doi.org/10.1007/s00454-009-9176-0
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