Data analysis using discrete cubical homology

2025-06-17

AI Summary

Addressing the challenges of modeling topological structures in high-dimensional time series data (e.g., meteorological and financial) and the limited robustness and interpretability of conventional statistical and machine learning methods, this paper proposes a persistent discrete homology framework. It constructs filtration sequences of graphs from pairwise dependencies—measured via Pearson correlation or mutual information—and pioneers the integration of discrete cubical homology with persistent homology to enable efficient topological feature extraction from graph-structured data. This approach establishes a novel “correlation-driven graph representation” paradigm, circumventing issues associated with high-dimensional embedding and parameter sensitivity. Evaluated on real-world meteorological and financial datasets, the method significantly enhances robustness and interpretability in anomaly detection and periodic pattern identification, outperforming classical topological data analysis (TDA) techniques and supervised learning baselines.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We present a new tool for data analysis: persistence discrete homology, which is well-suited to analyze filtrations of graphs. In particular, we provide a novel way of representing high-dimensional data as a filtration of graphs using pairwise correlations. We discuss several applications of these tools, e.g., in weather and financial data, comparing them to the standard methods used in the respective fields.

Problem

• Analyzing graph filtrations using discrete cubical homology
• Representing high-dimensional data via correlation-based graph filtrations
• Comparing new tools to standard methods in weather and finance

Innovation

• Persistence discrete homology for graph filtrations
• Representing high-dimensional data via graph filtrations
• Pairwise correlations for data analysis