# Persistence Curves: A canonical framework for summarizing persistence diagrams

arXiv: https://arxiv.org/abs/1904.07768 · HTML: https://mlsanity.com/paper/1904.07768/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Apr 2019.

## Built on by (most cited first, 30 of 33)

- [ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery](https://mlsanity.com/paper/2211.03808/) (2022)
- [Euler Characteristic Curves and Profiles: a stable shape invariant for big data problems](https://mlsanity.com/paper/2212.01666/) (2022)
- [A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis](https://mlsanity.com/paper/2109.00530/) (2021)
- [EMP: Effective Multidimensional Persistence for Graph Representation Learning](https://mlsanity.com/paper/2401.13713/) (2024)
- [A computationally efficient framework for vector representation of persistence diagrams](https://mlsanity.com/paper/2109.08239/) (2021)
- [Classification of Stochastic Processes with Topological Data Analysis](https://mlsanity.com/paper/2206.03973/) (2022)
- [A Topological Data Analysis Based Classifier](https://mlsanity.com/paper/2111.05214/) (2021)
- [Multiparameter Persistent Homology for Molecular Property Prediction](https://mlsanity.com/paper/2311.10808/) (2023)
- [Time-Aware Knowledge Representations of Dynamic Objects with Multidimensional Persistence](https://mlsanity.com/paper/2401.13157/) (2024)
- [Smart Vectorizations for Single and Multiparameter Persistence](https://mlsanity.com/paper/2104.04787/) (2021)
- [A fast topological approach for predicting anomalies in time-varying graphs](https://mlsanity.com/paper/2305.06523/) (2023)
- [Vector Summaries of Persistence Diagrams for Permutation-based Hypothesis Testing](https://mlsanity.com/paper/2306.06257/) (2023)
- [A Topological Machine Learning Pipeline for Classification](https://mlsanity.com/paper/2309.15276/) (2023)
- [SE(3)-Invariant Multiparameter Persistent Homology for Chiral-Sensitive Molecular Property Prediction](https://mlsanity.com/paper/2312.07633/) (2023)
- [Time topological analysis of EEG using signature theory](https://mlsanity.com/paper/2404.15328/) (2024)
- [Comparing the Effects of Persistence Barcodes Aggregation and Feature Concatenation on Medical Imaging](https://mlsanity.com/paper/2505.23637/) (2025)
- [CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizations](https://mlsanity.com/paper/2510.12795/) (2025)
- [Hierarchical Persistence Velocity for Network Anomaly Detection: Theory and Applications to Cryptocurrency Markets](https://mlsanity.com/paper/2512.14615/) (2025)
- [Topology-Aware Multiscale Mixture of Experts for Efficient Molecular Property Prediction](https://mlsanity.com/paper/2601.12637/) (2026)
- [From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Features](https://mlsanity.com/paper/2606.11911/) (2026)
- [Individual Topology Structure of Eye Movement Trajectories](https://mlsanity.com/paper/2205.10667/) (2022)
- [Prediction of textrmCO2 Adsorption in Nano-Pores with Graph Neural Networks](https://mlsanity.com/paper/2209.07567/) (2022)
- [Topology-enhanced machine learning for speech signal processing](https://mlsanity.com/paper/2311.15210/) (2023)
- [A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams](https://mlsanity.com/paper/2402.14489/) (2024)
- [A Vectorization Method Induced By Maximal Margin Classification For Persistent Diagrams](https://mlsanity.com/paper/2407.21298/) (2024)
- [Sleep Staging from Airflow Signals Using Fourier Approximations of Persistence Curves](https://mlsanity.com/paper/2411.07964/) (2024)
- [Vectorization of Persistence Diagrams for Topological Data Analysis in R and Python Using TDAvec Package](https://mlsanity.com/paper/2411.17340/) (2024)
- [Classification of Firn Data via Topological Features](https://mlsanity.com/paper/2504.16150/) (2025)
- [A Novel Patch-Based TDA Approach for Computed Tomography Imaging](https://mlsanity.com/paper/2512.12108/) (2025)
- [G-LoG Bi-filtration for Medical Image Classification](https://mlsanity.com/paper/2602.18329/) (2026)

## Abstract (arXiv)

Persistence diagrams are one of the main tools in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to be challenging as the space lacks an inner product. For that reason, transforming these diagrams in a way that is compatible with machine learning is an important topic currently researched in TDA. In this paper, our main contribution consists of three components. First, we develop a general and unifying framework of vectorizing diagrams that we call the textitPersistence Curves (PCs), and show that several well-known summaries, such as Persistence Landscapes, fall under the PC framework. Second, we propose several new summaries based on PC framework and provide a theoretical foundation for their stability analysis. Finally, we apply proposed PCs to two applications---texture classification and determining the parameters of a discrete dynamical system; their performances are competitive with other TDA methods.