# A Framework for Differential Calculus on Persistence Barcodes

arXiv: https://arxiv.org/abs/1910.00960 · HTML: https://mlsanity.com/paper/1910.00960/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Oct 2019.

## Built on by (most cited first, 29 of 29)

- [Learning Topology-Preserving Data Representations](https://mlsanity.com/paper/2302.00136/) (2023)
- [Going beyond persistent homology using persistent homology](https://mlsanity.com/paper/2311.06152/) (2023)
- [Persistent Homology with Improved Locality Information for more Effective Delineation](https://mlsanity.com/paper/2110.06295/) (2021)
- [Predicting the generalization gap in neural networks using topological data analysis](https://mlsanity.com/paper/2203.12330/) (2022)
- [Diffeomorphic interpolation for efficient persistence-based topological optimization](https://mlsanity.com/paper/2405.18820/) (2024)
- [A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis](https://mlsanity.com/paper/2109.00530/) (2021)
- [Topological Node2vec: Enhanced Graph Embedding via Persistent Homology](https://mlsanity.com/paper/2309.08241/) (2023)
- [D-GRIL: End-to-End Topological Learning with 2-parameter Persistence](https://mlsanity.com/paper/2406.07100/) (2024)
- [Disentanglement Learning via Topology](https://mlsanity.com/paper/2308.12696/) (2023)
- [Topology-Preserving Dimensionality Reduction via Interleaving Optimization](https://mlsanity.com/paper/2201.13012/) (2022)
- [Optimisation of Spectral Wavelets for Persistence-based Graph Classification](https://mlsanity.com/paper/2101.05201/) (2021)
- [Scale-Free Image Keypoints Using Differentiable Persistent Homology](https://mlsanity.com/paper/2406.01315/) (2024)
- [T-REGS: Minimum Spanning Tree Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2510.23484/) (2025)
- [Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraints](https://mlsanity.com/paper/2601.11409/) (2026)
- [Reconstruction of univariate functions from directional persistence diagrams](https://mlsanity.com/paper/2203.01894/) (2022)
- [Filtration learning in exact multi-parameter persistent homology and classification of time-series data](https://mlsanity.com/paper/2406.19587/) (2024)
- [Topological Approach for Data Assimilation](https://mlsanity.com/paper/2411.18627/) (2024)
- [Towards Scalable Topological Regularizers](https://mlsanity.com/paper/2501.14641/) (2025)
- [Cover Learning for Large-Scale Topology Representation](https://mlsanity.com/paper/2503.09767/) (2025)
- [Topological Regularization for Dense Prediction](https://mlsanity.com/paper/2111.10984/) (2021)
- [Decorrelating neurons using persistence](https://mlsanity.com/paper/2308.04870/) (2023)
- [Hall Effect Thruster Forecasting using a Topological Approach for Data Assimilation](https://mlsanity.com/paper/2504.06157/) (2025)
- [Dynamical System Parameter Path Optimization using Persistent Homology](https://mlsanity.com/paper/2505.00782/) (2025)
- [Cycles Communities from the Perspective of Dendrograms and Gradient Sampling](https://mlsanity.com/paper/2512.12974/) (2025)
- [TopoFisher: Learning Topological Summary Statistics by Maximizing Fisher Information](https://mlsanity.com/paper/2605.07720/) (2026)
- [PFlow-T: A Persistence-Driven Forward Process for Topology-Controlled Generation](https://mlsanity.com/paper/2605.17555/) (2026)
- [Non-negative Matrix Factorisation with Topological Regularisation](https://mlsanity.com/paper/2606.17531/) (2026)
- [COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules](https://mlsanity.com/paper/2609.22012/) (2026)
- [A differentiability framework for zigzag persistent homology via linear interpolation](https://mlsanity.com/paper/2609.39242/) (2026)

## 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.