Persistent Stanley--Reisner Theory

TL;DR

This work introduces persistent Stanley-Reisner theory (PSRT), extending classical Stanley-Reisner algebra to filtrations and defining persistent invariants such as $\beta_{i,i+j}^{t,t'}$, $h_m^{t,t'}$, and $f_{m-1}^{t,t'}$, along with persistent facet ideals and facet persistence modules. By extending Hochster's formula to the filtration setting, PSRT links algebraic invariants to persistence across scales, yielding facet persistence barcodes that track births and deaths of facet primes. The authors prove stability results ensuring small changes in the filtration induce small changes in the facet persistence diagrams, mirroring robustness in persistent homology. They validate the approach on molecular data, demonstrating discrimination of isomeric structures and high-accuracy classification of metal halide perovskites using purely geometric information, outperforming baseline persistent Betti-number methods.

Abstract

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.

Figures (8)

• Figure 1: A geometric representation of six vertices forming a pyramid with two distinct loops, each attached to different edges.
• Figure 2: Three representative plots of the filtration of a triangular bipyramid with an equatorial triangular cross-section.
• Figure 3: An illustration of the persistent variations of the $h$-vectors, $f$-vectors, Betti numbers, and graded Betti numbers, respectively. Each curve represents a function that assumes discrete integer values. To enhance the visual clarity of the plots and prevent overlap, a slight increment is added to the curves, ensuring that each curve remains distinguishable within the graphical representation.
• Figure 4: Illustration of the methodology and key steps in the proposed application of persistent Stanley--Reisner theory. Given a molecular input, a corresponding simplicial complex with an associated filtration is generated. Critical values and facet persistence barcodes are computed, leading to the construction of relevant features.
• Figure 5: Structural representations and filtrations of two isomers. The top row presents the first isomer and its corresponding filtration, while the bottom row depicts the second and filtration processes. In both structures, hydrogen atoms are represented in white, boron atoms in green, and carbon atoms in brown.

Theorems & Definitions (24)

• Theorem 2.1: Hochster's Formula
• Example 2.2
• Definition 3.1: Persistent Homology Groups
• Definition 3.2: Persistent Stanley--Reisner Graded Betti Numbers
• Lemma 3.3: Graded Free Resolution and Hilbert Series
• Example 3.4
• Definition 3.5: Prime Monomial Ideals
• Definition 3.6: Stanley--Reisner Critical Value
• Lemma 3.7: Critical Value Lemma
• proof