mlsanityData to 7 Oct 2026

A Framework for Differential Calculus on Persistence Barcodes

arXiv 1910.00960 ↗First posted 2 Oct 2019math.AT cs.CG math.DGcited by 29

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Abstract (arXiv)

We define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be computed. The two derived notions of differentiability (respectively from and to the space of barcodes) combine together naturally to produce a chain rule that enables the use of gradient descent for objective functions factoring through the space of barcodes. We illustrate the versatility of this framework by showing how it can be used to analyze the smoothness of various parametrized families of filtrations arising in topological data analysis.

Key facts. Cited by 29 papers in the map (data through 7 Oct 2026). First posted on arXiv 2 Oct 2019.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 4; 2022: 3; 2023: 5; 2024: 5; 2025: 6; 2026: 6.

29 papers.

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  1. 2023Learning Topology-Preserving Data Representationscited by 30
  2. 2023Going beyond persistent homology using persistent homologycited by 21
  3. 2021Persistent Homology with Improved Locality Information for more Effective Delineationcited by 10
  4. 2022Predicting the generalization gap in neural networks using topological data analysiscited by 9
  5. 2024Diffeomorphic interpolation for efficient persistence-based topological optimizationcited by 6
  6. 2021A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysiscited by 5
  7. 2023Topological Node2vec: Enhanced Graph Embedding via Persistent Homologycited by 5
  8. 2024D-GRIL: End-to-End Topological Learning with 2-parameter Persistencecited by 5
  9. 2023Disentanglement Learning via Topologycited by 4
  10. 2022Topology-Preserving Dimensionality Reduction via Interleaving Optimizationcited by 3
  11. 2021Optimisation of Spectral Wavelets for Persistence-based Graph Classificationcited by 2
  12. 2024Scale-Free Image Keypoints Using Differentiable Persistent Homologycited by 2
17 more
  1. 2025T-REGS: Minimum Spanning Tree Regularization for Self-Supervised Learningcited by 2
  2. 2026Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraintscited by 2
  3. 2022Reconstruction of univariate functions from directional persistence diagramscited by 1
  4. 2024Topological Approach for Data Assimilationcited by 1
  5. 2025Towards Scalable Topological Regularizerscited by 1
  6. 2025Cover Learning for Large-Scale Topology Representationcited by 1
  7. 2021Topological Regularization for Dense Predictioncited by 0
  8. 2023Decorrelating neurons using persistencecited by 0
  9. 2024Filtration learning in exact multi-parameter persistent homology and classification of time-series datacited by 0
  10. 2025Hall Effect Thruster Forecasting using a Topological Approach for Data Assimilationcited by 0
  11. 2025Dynamical System Parameter Path Optimization using Persistent Homologycited by 0
  12. 2025Cycles Communities from the Perspective of Dendrograms and Gradient Samplingcited by 0
  13. 2026TopoFisher: Learning Topological Summary Statistics by Maximizing Fisher Informationcited by 0
  14. 2026PFlow-T: A Persistence-Driven Forward Process for Topology-Controlled Generationcited by 0
  15. 2026Non-negative Matrix Factorisation with Topological Regularisationcited by 0
  16. 2026COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modulescited by 0
  17. 2026A differentiability framework for zigzag persistent homology via linear interpolationcited by 0