The Density of Cross-Persistence Diagrams and Its Applications

2026-03-12

AI Summary

Traditional persistence diagrams struggle to capture the interactive topological relationships between point clouds and lack cross-structural modeling capacity. This work establishes, for the first time, the existence and statistical foundations of cross-persistence diagram densities and introduces an end-to-end framework that integrates topological data analysis, statistical learning, and deep learning to directly predict these densities from point cloud coordinates and distance matrices. A novel noise-augmentation mechanism is innovatively incorporated to enhance the discriminative power of point clouds, significantly extending the applicability of topological data analysis in cross-structural settings. Experiments demonstrate that the proposed method achieves state-of-the-art performance in both density prediction and point cloud discrimination across multiple datasets, while also showing promising potential in geometric analyses of time series and AI-generated text.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• 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) provides powerful tools to explore the shape and structure of data through topological features such as clusters, loops, and voids. Persistence diagrams are a cornerstone of TDA, capturing the evolution of these features across scales. While effective for analyzing individual manifolds, persistence diagrams do not account for interactions between pairs of them. Cross-persistence diagrams (cross-barcodes), introduced recently, address this limitation by characterizing relationships between topological features of two point clouds. In this work, we present the first systematic study of the density of cross-persistence diagrams. We prove its existence, establish theoretical foundations for its statistical use, and design the first machine learning framework for predicting cross-persistence density directly from point cloud coordinates and distance matrices. Our statistical approach enables the distinction of point clouds sampled from different manifolds by leveraging the linear characteristics of cross-persistence diagrams. Interestingly, we find that introducing noise can enhance our ability to distinguish point clouds, uncovering its novel utility in TDA applications. We demonstrate the effectiveness of our methods through experiments on diverse datasets, where our approach consistently outperforms existing techniques in density prediction and achieves superior results in point cloud distinction tasks. Our findings contribute to a broader understanding of cross-persistence diagrams and open new avenues for their application in data analysis, including potential insights into time-series domain tasks and the geometry of AI-generated texts. Our code is publicly available at https://github.com/Verdangeta/TDA_experiments

Problem

• cross-persistence diagrams
• topological data analysis
• point cloud comparison
• density estimation
• topological features

Innovation

• cross-persistence diagrams
• topological data analysis
• density estimation
• point cloud classification
• noise-enhanced discrimination