# Learning on Persistence Diagrams as Radon Measures

arXiv: https://arxiv.org/abs/2212.08295 · HTML: https://mlsanity.com/paper/2212.08295/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 18 references that are in the map.
- First posted on arXiv 16 Dec 2022.

## Builds on (most cited first, 7 of 7)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Understanding the Topology and the Geometry of the Space of Persistence Diagrams via Optimal Partial Transport](https://mlsanity.com/paper/1901.03048/) (2019)
- [The density of expected persistence diagrams and its kernel based estimation](https://mlsanity.com/paper/1802.10457/) (2018)
- [Approximating Continuous Functions on Persistence Diagrams Using Template Functions](https://mlsanity.com/paper/1902.07190/) (2019)
- [Combining crowd-sourcing and deep learning to explore the meso-scale organization of shallow convection](https://mlsanity.com/paper/1906.01906/) (2019)
- [A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science](https://mlsanity.com/paper/2207.10552/) (2022)
- [Adaptive template systems: Data-driven feature selection for learning with persistence diagrams](https://mlsanity.com/paper/1910.06741/) (2019)

## Abstract (arXiv)

Persistence diagrams are common descriptors of the topological structure of data appearing in various classification and regression tasks. They can be generalized to Radon measures supported on the birth-death plane and endowed with an optimal transport distance. Examples of such measures are expectations of probability distributions on the space of persistence diagrams. In this paper, we develop methods for approximating continuous functions on the space of Radon measures supported on the birth-death plane, as well as their utilization in supervised learning tasks. Indeed, we show that any continuous function defined on a compact subset of the space of such measures (e.g., a classifier or regressor) can be approximated arbitrarily well by polynomial combinations of features computed using a continuous compactly supported function on the birth-death plane (a template). We provide insights into the structure of relatively compact subsets of the space of Radon measures, and test our approximation methodology on various data sets and supervised learning tasks.