Persistent Stanley--Reisner Theory

Topological data analysis (TDA) has emerged as an effective approach in data science, with its key technique, persistent homology, rooted in algebraic topology. Although alternative approaches based on differential topology, geometric topology, and combinatorial Laplacians have been proposed, combinatorial commutative algebra has hardly been developed for machine learning and data science. In this work, we introduce persistent Stanley-Reisner theory to bridge commutative algebra, combinatorial algebraic topology, machine learning, and data science. We propose persistent h-vectors, persistent f-vectors, persistent graded Betti numbers, persistent facet ideals, and facet persistence modules. Stability analysis indicates that these algebraic invariants are stable against geometric perturbations. We employ a machine learning prediction on a molecular dataset to demonstrate the utility of the proposed persistent Stanley-Reisner theory for practical applications.