The Theory of Multidimensional Persistence

@article{Carlsson2007TheTO,
  title={The Theory of Multidimensional Persistence},
  author={Gunnar E. Carlsson and Afra Zomorodian},
  journal={Discrete \& Computational Geometry},
  year={2007},
  volume={42},
  pages={71-93}
}
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… 
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