# Approximating Continuous Functions on Persistence Diagrams Using Template Functions

arXiv: https://arxiv.org/abs/1902.07190 · HTML: https://mlsanity.com/paper/1902.07190/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Feb 2019.

## Built on by (most cited first, 10 of 10)

- [Data-driven and Automatic Surface Texture Analysis Using Persistent Homology](https://mlsanity.com/paper/2110.10005/) (2021)
- [Transfer Learning for Autonomous Chatter Detection in Machining](https://mlsanity.com/paper/2204.05400/) (2022)
- [A Pipeline for Data-Driven Learning of Topological Features with Applications to Protein Stability Prediction](https://mlsanity.com/paper/2408.04847/) (2024)
- [Learning on Persistence Diagrams as Radon Measures](https://mlsanity.com/paper/2212.08295/) (2022)
- [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)
- [Persistence-Augmented Neural Networks](https://mlsanity.com/paper/2604.08469/) (2026)
- [TopoAgent: An Agentic Framework for Automated Topology Learning in Medical Imaging](https://mlsanity.com/paper/2606.29763/) (2026)
- [Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams](https://mlsanity.com/paper/2607.27126/) (2026)
- [Persistent Cross Entropy](https://mlsanity.com/paper/2608.24549/) (2026)

## Abstract (arXiv)

The persistence diagram is an increasingly useful tool from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. The most success to date has come from methods that map persistence diagrams into vector spaces, in a way which maximizes the structure preserved. This process is commonly referred to as featurization. In this paper, we describe a mathematical framework for featurization called template functions, and we show that it addresses the problem of approximating continuous functions on compact subsets of the space of persistence diagrams. Specifically, we begin by characterizing relative compactness with respect to the bottleneck distance, and then provide explicit theoretical methods for constructing compact-open dense subsets of continuous functions on persistence diagrams. These dense subsets -- obtained via template functions -- are leveraged for supervised learning tasks with persistence diagrams. Specifically, we test the method for classification and regression algorithms on several examples including shape data and dynamical systems.