Rank invariant for Zigzag Modules

10/26/2018

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by   Woojin Kim, et al.

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The rank invariant is of great interest in studying both one-dimensional and multidimensional persistence modules. Based on a category theoretical point of view, we generalize the rank invariant to zigzag modules. We prove that the rank invariant of a zigzag module recovers its interval decomposition. This proves that the rank invariant is a complete invariant for zigzag modules. The degree of difference between the rank invariants of any two zigzag modules can be measured via the erosion distance proposed by A. Patel. We show that this erosion distance is bounded from above by the interleaving distance between the zigzag modules up to a multiplicative constant. Our construction allows us to extend the notion of generalized persistence diagram by A. Patel to zigzag modules valued in a symmetric monoidal bicomplete category.