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Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets
Ángel Javier Alonso1, Michael Kerber1
1Institute of Geometry, Graz University of Technology, Graz, Austria.
Discrete & Computational Geometry
|December 1, 2025
Summary
We introduce rooted subsets to analyze persistence modules, simplifying the decomposition of zero-dimensional modules. This method reveals that at least 25% of summands in density-Rips filtrations are interval modules.
Area of Science:
- Topological Data Analysis
- Algebraic Topology
- Computational Geometry
Background:
- Persistence modules are fundamental in TDA for analyzing data shape.
- Decomposition of these modules is crucial for extracting meaningful topological features.
- Existing methods can be computationally intensive for large datasets.
Purpose of the Study:
- To develop a combinatorial approach for decomposing zero-dimensional persistence modules.
- To introduce the concept of rooted subsets and its relation to persistence modules.
- To provide a lower bound on the number of interval modules in specific filtrations.
Main Methods:
- Studying decomposition at the set level before vector spaces.
- Defining and utilizing the combinatorial notion of rooted subsets.
- Relating rooted subsets to clustering behavior in filtered metric spaces.
- Generalizing the elder rule and connecting to constant conqueror.
Main Results:
- Rooted subsets provide a link between metric space clustering and persistence module decomposition.
- Efficient identification of intervals within persistence module decompositions is enabled.
- A lower bound of 25% for interval modules in density-Rips filtrations is established.
- Rooted subsets generalize existing combinatorial rules.
Conclusions:
- The rooted subset approach offers an efficient and insightful method for persistence module analysis.
- This work provides theoretical guarantees on the structure of persistence modules from geometric data.
- The findings have implications for understanding the complexity and interpretability of topological features.
Keywords:
ClusteringDecomposition of persistence modulesElder ruleMultiparameter persistence homologyMore Related Videos
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