# Understanding the Topology and the Geometry of the Space of Persistence Diagrams via Optimal Partial Transport

arXiv: https://arxiv.org/abs/1901.03048 · HTML: https://mlsanity.com/paper/1901.03048/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2019.

## Built on by (most cited first, 17 of 17)

- [Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures](https://mlsanity.com/paper/2306.03801/) (2023)
- [Estimation and Quantization of Expected Persistence Diagrams](https://mlsanity.com/paper/2105.04852/) (2021)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [A novel approach for wafer defect pattern classification based on topological data analysis](https://mlsanity.com/paper/2209.08945/) (2022)
- [Topological Node2vec: Enhanced Graph Embedding via Persistent Homology](https://mlsanity.com/paper/2309.08241/) (2023)
- [Topological Information Retrieval with Dilation-Invariant Bottleneck Comparative Measures](https://mlsanity.com/paper/2104.01672/) (2021)
- [An Homogeneous Unbalanced Regularized Optimal Transport model with applications to Optimal Transport with Boundary](https://mlsanity.com/paper/2201.02082/) (2022)
- [k-Means Clustering for Persistent Homology](https://mlsanity.com/paper/2210.10003/) (2022)
- [Topological data analysis of human vowels: Persistent homologies across representation spaces](https://mlsanity.com/paper/2310.06508/) (2023)
- [Discrete transforms of quantized persistence diagrams](https://mlsanity.com/paper/2312.17093/) (2023)
- [Nonparametric regression on random geometric graphs sampled from submanifolds](https://mlsanity.com/paper/2405.20909/) (2024)
- [Towards Scalable Topological Regularizers](https://mlsanity.com/paper/2501.14641/) (2025)
- [Monitoring weeder robots and anticipating their functioning by using advanced topological data analysis](https://mlsanity.com/paper/2108.08570/) (2021)
- [Learning on Persistence Diagrams as Radon Measures](https://mlsanity.com/paper/2212.08295/) (2022)
- [Hierarchical Clustering in LambdaCDM Cosmologies via Persistence Energy](https://mlsanity.com/paper/2401.01988/) (2024)
- [Persistence Spheres: Bi-continuous Representations of Persistence Diagrams](https://mlsanity.com/paper/2509.16999/) (2025)
- [Revisiting the Sliced Wasserstein Kernel for persistence diagrams: a Figalli-Gigli approach](https://mlsanity.com/paper/2602.06539/) (2026)

## Abstract (arXiv)

Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come. In this article, by considering the space of persistence diagrams as a space of discrete measures, and by observing that its metrics can be expressed as optimal partial transport problems, we introduce a generalization of persistence diagrams, namely Radon measures supported on the upper half plane. Such measures naturally appear in topological data analysis when considering continuous representations of persistence diagrams (e.g. persistence surfaces) but also as limits for laws of large numbers on persistence diagrams or as expectations of probability distributions on the persistence diagrams space. We explore topological properties of this new space, which will also hold for the closed subspace of persistence diagrams. New results include a characterization of convergence with respect to Wasserstein metrics, a geometric description of barycenters (Fr'echet means) for any distribution of diagrams, and an exhaustive description of continuous linear representations of persistence diagrams. We also showcase the strength of this framework to study random persistence diagrams by providing several statistical results made meaningful thanks to this new formalism.