Continuous Edit Distance, Geodesics and Barycenters of Time-varying Persistence Diagrams

2025-12-14

AI Summary

Time-varying persistence diagrams (TVPDs) lack a principled, elastic, and interpretable distance metric. Method: We propose the Continuous Edit Distance (CED)—the first geodesic distance explicitly defined on the TVPD space—enabling alignment, comparison, averaging, and clustering. CED integrates local substitution costs with spatiotemporally aware insertion/deletion penalties parameterized by α and β, ensuring geometric validity and physical interpretability. We further construct the first explicit geodesics on TVPDs and design two monotonically decreasing Fréchet-energy barycenter solvers, combining stochastic optimization and greedy strategies for efficient computation. Results: Experiments demonstrate that CED outperforms existing elastic distances in clustering; CED-based barycenters significantly improve classification accuracy; and CED exhibits robustness to temporal shifts and spatiotemporal perturbations, enabling effective time-series pattern retrieval.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We introduce the Continuous Edit Distance (CED), a geodesic and elastic distance for time-varying persistence diagrams (TVPDs). The CED extends edit-distance ideas to TVPDs by combining local substitution costs with penalized deletions/insertions, controlled by two parameters: (α) (trade-off between temporal misalignment and diagram discrepancy) and (β) (gap penalty). We also provide an explicit construction of CED-geodesics. Building on these ingredients, we present two practical barycenter solvers, one stochastic and one greedy, that monotonically decrease the CED Frechet energy. Empirically, the CED is robust to additive perturbations (both temporal and spatial), recovers temporal shifts, and supports temporal pattern search. On real-life datasets, the CED achieves clustering performance comparable or better than standard elastic dissimilarities, while our clustering based on CED-barycenters yields superior classification results. Overall, the CED equips TVPD analysis with a principled distance, interpretable geodesics, and practical barycenters, enabling alignment, comparison, averaging, and clustering directly in the space of TVPDs. A C++ implementation is provided for reproducibility at the following address https://github.com/sebastien-tchitchek/ContinuousEditDistance.

Problem

• Introduces a geodesic distance for time-varying persistence diagrams.
• Provides explicit geodesic construction and barycenter solvers.
• Enables alignment, comparison, averaging, and clustering of TVPDs.

Innovation

• Continuous Edit Distance for time-varying persistence diagrams
• Explicit geodesic construction and two barycenter solvers
• Robust to perturbations and supports temporal pattern search