Commutative algebra-enhanced topological data analysis

TL;DR

This work addresses the limitations of persistent homology in Topological Data Analysis by introducing two algebraic frameworks that reveal finer topology and combinatorics. It develops persistent ideals derived from edge and Stanley–Reisner ideals, providing new algebraic persistence barcodes via associated primes that can recover and augment traditional persistence information. It also introduces a persistent chain complex of free modules labeled by a UFD, with evaluation/localization results showing equivalence to the standard chain complex and enabling local topological insight, including a graded version for $R=\mathbb{F}[x_1,\dots,x_t]$. Collectively, these approaches fuse computational topology with commutative algebra, yielding richer invariants and a unified algebraic-geometric perspective for analyzing filtrations and their persistence in data.

Abstract

Topological Data Analysis (TDA) combines computational topology and data science to extract and analyze intrinsic topological and geometric structures in data set in a metric space. While the persistent homology (PH), a widely used tool in TDA, which tracks the lifespan information of topological features through a filtration process, has shown its effectiveness in applications,it is inherently limited in homotopy invariants and overlooks finer geometric and combinatorial details. To bridge this gap, we introduce two novel commutative algebra-based frameworks which extend beyond homology by incorporating tools from computational commutative algebra : (1) \emph{the persistent ideals} derived from the decomposition of algebraic objects associated to simplicial complexes, like those in theory of edge ideals and Stanley--Reisner ideals, which will provide new commutative algebra-based barcodes and offer a richer characterization of topological and geometric structures in filtrations.(2)\emph{persistent chain complex of free modules} associated with traditional persistent simplicial complex by labelling each chain in the chain complex of the persistent simplicial complex with elements in a commutative ring, which will enable us to detect local information of the topology via some pure algebraic operations. \emph{Crucially, both of the two newly-established framework can recover topological information got from conventional PH and will give us more information.} Therefore, they provide new insights in computational topology, computational algebra and data science.

Figures (2)

• Figure 1: Illustration of an example distance matrix for a point cloud consisting of four points indexed by 1, 2, 3, and 4, along with the corresponding persistence barcodes (PBs) induced by the persistent Stanley--Reisner (SR) ideals and conventional persistent homology. Both PBs are derived from the same Vietoris--Rips (VR) filtration constructed over the point cloud. Each persistence interval in the PB of the persistent SR ideals corresponds to a monomial ideal generated by a subset of $\{ x_i \mid i = 1, 2, 3, 4 \}$, which is annotated by black text next to the corresponding interval. Equivalently, as indicated by the red annotations, each monomial ideal corresponds to a maximal simplex in the simplicial complex associated with a fixed radius in the VR filtration, encoding the lifespan information of maximal simplices of various dimensions within the VR filtration.
• Figure 2: Figures for the simplicial complexes in Example \ref{['eg: relation between edge ideals and SR ideals']}

Theorems & Definitions (56)

• Example 2.1: Clique complexes
• Remark 2.2
• Theorem 2.3
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Corollary 3.3
• proof
• Lemma 3.4