Commutative algebra-enhanced topological data analysis

2025-04-12

AI Summary

Persistent homology (PH), a cornerstone of topological data analysis, suffers from inherent limitations in capturing homotopy invariance, geometric detail, and combinatorial structure. To address these, we introduce a novel algebraic framework grounded in commutative algebra, built upon two key constructs: *persistent ideals* and *ring-labeled free-module chain complexes*. This framework embeds filtered data into algebraic structures over polynomial rings. It recovers classical PH barcodes while generating new *algebraic barcodes*—multi-scale, fine-grained, and locally defined algebraic representations of topological features. By integrating Stanley–Reisner ideals, edge ideal decompositions, and persistent filtration modeling, our approach extends the algebraic foundations of PH theoretically and enhances its capacity to jointly encode geometric and combinatorial data structure. The framework thus fosters deeper synergy among computational topology, commutative algebra, and data science.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Other Foundations of Data Mining & Knowledge Management
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Extend persistent homology with commutative algebra for richer topological analysis
• Develop persistent ideals to capture finer geometric and combinatorial details
• Construct persistent chain complexes to detect local topological information

Innovation

• Persistent ideals from algebraic decomposition
• Persistent chain complex with commutative rings
• Recovers and extends persistent homology information