Persistent Stanley-Reisner Theory

Area of Science:

• Algebraic Topology
• Machine Learning
• Data Science

Background:

• Topological Data Analysis (TDA) utilizes persistent homology from algebraic topology.
• Existing TDA methods lack development in combinatorial commutative algebra for machine learning.
• Bridging this gap is crucial for advancing data science techniques.

Purpose of the Study:

• Introduce persistent Stanley-Reisner theory.
• Connect combinatorial commutative algebra with TDA, machine learning, and data science.
• Develop new algebraic invariants for data analysis.

Main Methods:

• Developed persistent h-vectors, persistent f-vectors, and persistent graded Betti numbers.
• Introduced persistent facet ideals and facet persistence modules.
• Conducted stability analysis to ensure robustness against geometric perturbations.

Main Results:

• Demonstrated the stability of proposed algebraic invariants.
• Successfully applied persistent Stanley-Reisner theory to a molecular dataset.
• Showcased the utility of the new theory in machine learning predictions.

Conclusions:

• Persistent Stanley-Reisner theory effectively bridges commutative algebra and data science.
• The proposed invariants offer a robust framework for topological data analysis.
• This work opens new avenues for applying algebraic methods in machine learning.