Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets.

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.