Computing Generalized Ranks of Persistence Modules via Unfolding to Zigzag Modules

2024-03-12

arXiv.org

AI Summary

Computing the generalized rank of persistence modules indexed by arbitrary posets is challenging—especially in multiparameter settings where natural boundary structures are absent, precluding direct application of zigzag persistence algorithms. Method: We introduce a systematic unfolding-and-folding framework that transforms any poset-indexed module into an equivalent zigzag module, then leverages categorical limits and colimits to identify fully collapsible interval modules for exact generalized rank extraction. Contributions: (i) First rigorous criterion for poset-to-zigzag unfolding; (ii) Generalization beyond 1D and 2D settings to arbitrary dimension $d$ and general poset structures; (iii) First linear-time algorithm for 1D homology; (iv) Complexity improvement for $d$-th homology of $d$-complexes. Our approach unifies persistent homology, zigzag decomposition, and graph homology techniques to establish a rigorous computational paradigm for generalized rank.

Technology Category

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• Machine Learning: Matrix & Tensor Methods
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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

For a $P$-indexed persistence module ${sf M}$, the (generalized) rank of ${sf M}$ is defined as the rank of the limit-to-colimit map for the diagram of vector spaces of ${sf M}$ over the poset $P$. For $2$-parameter persistence modules, recently a zigzag persistence based algorithm has been proposed that takes advantage of the fact that generalized rank for $2$-parameter modules is equal to the number of full intervals in a zigzag module defined on the boundary of the poset. Analogous definition of boundary for $d$-parameter persistence modules or general $P$-indexed persistence modules does not seem plausible. To overcome this difficulty, we first unfold a given $P$-indexed module ${sf M}$ into a zigzag module ${sf M}_{ZZ}$ and then check how many full interval modules in a decomposition of ${sf M}_{ZZ}$ can be folded back to remain full in a decomposition of ${sf M}$. This number determines the generalized rank of ${sf M}$. For special cases of degree-$d$ homology for $d$-complexes, we obtain a more efficient algorithm including a linear time algorithm for degree-$1$ homology in graphs.

Problem

• Computing generalized ranks of persistence modules
• Unfolding modules into zigzag persistence
• Efficient algorithms for homology in complexes

Innovation

• Unfold persistence modules into zigzag modules
• Count full intervals in decomposition via folding
• Linear time algorithm for degree-1 homology