A comparison framework for interleaved persistence modules.

Area of Science:

• Computational geometry
• Algebraic topology
• Data analysis

Background:

• The induced matching theorem and algebraic stability theorem are foundational in computational topology.
• Persistence modules are key structures in Topological Data Analysis (TDA).
• Current error bounds for persistence diagrams often rely on bottleneck distance.

Purpose of the Study:

• To generalize the induced matching theorem and the algebraic stability theorem for persistence modules.
• To develop a method for computing rigorous error bounds in persistence diagrams.
• To extend the applicability of stability theorems in TDA.

Main Methods:

• Generalization of the induced matching theorem.
• Proof of a generalized algebraic stability theorem for $$-indexed persistence modules.
• Illustrative examples demonstrating error bound computation.

Main Results:

• A novel generalization of the algebraic stability theorem for persistence modules.
• Demonstration of rigorous error bound computation in persistence diagrams.
• Error bounds that surpass the limitations of bottleneck distance.

Conclusions:

• The generalized algebraic stability theorem provides a powerful tool for TDA.
• This work enables more precise error quantification in persistence diagrams.
• The findings advance the theoretical and practical aspects of TDA.