The Density of Cross-Persistence Diagrams and Its Applications

• URL: http://arxiv.org/abs/2603.11623v1
• Date: Thu, 12 Mar 2026 07:33:15 GMT
• Title: The Density of Cross-Persistence Diagrams and Its Applications
• Abstract summary: Topological Data Analysis (TDA) provides powerful tools to explore the shape and structure of data through topological features such as clusters, loops, and voids.<n>Persistence diagrams are a cornerstone of TDA, capturing the evolution of these features across scales.<n>Cross-persistence diagrams (cross-barcodes), introduced recently, address this limitation by characterizing relationships between topological features of two point clouds.
• Score: 67.12329106585666
• License: http://creativecommons.org/licenses/by/4.0/
• 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

Related papers

• Connectivity Estimation using Stochastic Graph Heat Modelling [8.495508597640026]
  We extend noise-driven heat modelling on graphs for neurophysiological connectivity estimation.<n>We demonstrate that the technique is able to capture meaningful spatial structure across two experiments.<n>The explicit model formulation of our connectivity estimator has the potential to improve the interpretability of graph-based techniques.
  arXiv  Detail & Related papers  (2026-06-27T21:55:02Z)
• GeodesicNVS: Probability Density Geodesic Flow Matching for Novel View Synthesis [54.39598154430305]
  We propose a Data-to-Data Flow Matching framework that learns deterministic transformations directly between paired views.<n>PDG-FM constrains flow trajectories using geodesic interpolants derived from probability density metrics of pretrained diffusion models.<n>These results highlight the advantages of incorporating data-dependent geometric regularization into deterministic flow matching for consistent novel view generation.
  arXiv  Detail & Related papers  (2026-03-01T09:30:11Z)
• OWLEYE: Zero-Shot Learner for Cross-Domain Graph Data Anomaly Detection [48.77471686671269]
  OWLEYE is a novel framework that learns transferable patterns of normal behavior from multiple graphs.<n>We show that OWLEYE achieves superior performance and generalizability compared to state-of-the-art baselines.
  arXiv  Detail & Related papers  (2026-01-27T02:08:18Z)
• Persistent Homology-induced Graph Ensembles for Time Series Regressions [1.5728609542259502]
  We create an ensemble of Graph Neural Networks based on Persistent Homology filtration.<n>The ensemble aggregates the signals from the individual learners via an attention-based routing mechanism.<n>Four different real-world experiments on seismic activity prediction and traffic forecasting demonstrate that our approach consistently outperforms single-graph baselines.
  arXiv  Detail & Related papers  (2025-03-18T13:22:52Z)
• Deep End-to-End Survival Analysis with Temporal Consistency [49.77103348208835]
  We present a novel Survival Analysis algorithm designed to efficiently handle large-scale longitudinal data. A central idea in our method is temporal consistency, a hypothesis that past and future outcomes in the data evolve smoothly over time. Our framework uniquely incorporates temporal consistency into large datasets by providing a stable training signal.
  arXiv  Detail & Related papers  (2024-10-09T11:37:09Z)
• Discrete transforms of quantized persistence diagrams [0.5249805590164902]
  We introduce Qupid, a novel and simple method for vectorizing persistence diagrams. Key features are the choice of log-scaled grids that emphasize information contained near the diagonal in persistence diagrams. We conduct an in-depth experimental analysis of Qupid, showing that the simplicity of our method results in very low computational costs.
  arXiv  Detail & Related papers  (2023-12-28T16:11:11Z)
• Improving embedding of graphs with missing data by soft manifolds [51.425411400683565]
  The reliability of graph embeddings depends on how much the geometry of the continuous space matches the graph structure. We introduce a new class of manifold, named soft manifold, that can solve this situation. Using soft manifold for graph embedding, we can provide continuous spaces to pursue any task in data analysis over complex datasets.
  arXiv  Detail & Related papers  (2023-11-29T12:48:33Z)
• AMES: A Differentiable Embedding Space Selection Framework for Latent Graph Inference [6.115315198322837]
  We introduce the Attentional Multi-Embedding Selection (AMES) framework, a differentiable method for selecting the best embedding space for latent graph inference. Our framework consistently achieves comparable or superior results compared to previous methods for latent graph inference.
  arXiv  Detail & Related papers  (2023-11-20T16:24:23Z)
• $k$-Means Clustering for Persistent Homology [0.0]
  We prove convergence of the $k$-means clustering algorithm on persistence diagram space. We also establish theoretical properties of the solution to the optimization problem in the Karush--Kuhn--Tucker framework.
  arXiv  Detail & Related papers  (2022-10-18T17:18:51Z)
• GELATO: Geometrically Enriched Latent Model for Offline Reinforcement Learning [54.291331971813364]
  offline reinforcement learning approaches can be divided into proximal and uncertainty-aware methods. In this work, we demonstrate the benefit of combining the two in a latent variational model. Our proposed metrics measure both the quality of out of distribution samples as well as the discrepancy of examples in the data.
  arXiv  Detail & Related papers  (2021-02-22T19:42:40Z)
• Robust Persistence Diagrams using Reproducing Kernels [15.772439913138161]
  We develop a framework for constructing robust persistence diagrams from superlevel filtrations of robust density estimators constructed using kernels. We demonstrate the superiority of the proposed approach on benchmark datasets.
  arXiv  Detail & Related papers  (2020-06-17T17:16:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.