The Theory of Multidimensional Persistence |
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Authors: | Gunnar Carlsson Afra Zomorodian |
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Affiliation: | (1) Department of Mathematics, Stanford University, Stanford, CA, USA;(2) Department of Computer Science, Dartmouth College, Hanover, NH, USA |
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Abstract: | 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. The first author was partially supported by NSF under grant DMS-0354543. The second author was partially supported by DARPA under grant HR 0011-06-1-0038 and by ONR under grant N 00014-08-1-0908. Both authors were partially supported by DARPA under grant HR 0011-05-1-0007. |
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Keywords: | Computational topology Multidimensional analysis Persistent homology Persistence |
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