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

arXiv: https://arxiv.org/abs/2512.12939 · HTML: https://mlsanity.com/paper/2512.12939/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 75 references that are in the map.
- First posted on arXiv 15 Dec 2025.

## Builds on (most cited first, 19 of 19)

- [Soft-DTW: a Differentiable Loss Function for Time-Series](https://mlsanity.com/paper/1703.01541/) (2017)
- [The Topology ToolKit](https://mlsanity.com/paper/1805.09110/) (2018)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [Task-based Augmented Contour Trees with Fibonacci Heaps](https://mlsanity.com/paper/1902.04805/) (2019)
- [Progressive Wasserstein Barycenters of Persistence Diagrams](https://mlsanity.com/paper/1907.04565/) (2019)
- [Ranking Viscous Finger Simulations to an Acquired Ground Truth with Topology-aware Matchings](https://mlsanity.com/paper/1908.07841/) (2019)
- [TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data](https://mlsanity.com/paper/2009.01512/) (2020)
- [An Empirical Evaluation of Similarity Measures for Time Series Classification](https://mlsanity.com/paper/1401.3973/) (2014)
- [Persistence Atlas for Critical Point Variability in Ensembles](https://mlsanity.com/paper/1807.11212/) (2018)
- [Discrete Morse Sandwich: Fast Computation of Persistence Diagrams for Scalar Data -- An Algorithm and A Benchmark](https://mlsanity.com/paper/2206.13932/) (2022)
- [Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows -- An Experimental Study](https://mlsanity.com/paper/2207.14080/) (2022)
- [Nonsmooth Analysis and Subgradient Methods for Averaging in Dynamic Time Warping Spaces](https://mlsanity.com/paper/1701.06393/) (2017)
- [Topological data analysis of vortices in the magnetically-induced current density in LiH molecule](https://mlsanity.com/paper/2212.08690/) (2022)
- [Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2207.10960/) (2022)
- [Probabilistic Fr'echet Means for Time Varying Persistence Diagrams](https://mlsanity.com/paper/1307.6530/) (2013)
- [TTK is Getting MPI-Ready](https://mlsanity.com/paper/2310.08339/) (2023)
- [BondMatcher: H-Bond Stability Analysis in Molecular Systems](https://mlsanity.com/paper/2504.03205/) (2025)
- [Distributed Discrete Morse Sandwich: Efficient Computation of Persistence Diagrams for Massive Scalar Data](https://mlsanity.com/paper/2505.21266/) (2025)

## Abstract (arXiv)

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.