# The density of expected persistence diagrams and its kernel based estimation

arXiv: https://arxiv.org/abs/1802.10457 · HTML: https://mlsanity.com/paper/1802.10457/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2018.

## Built on by (most cited first, 13 of 13)

- [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)
- [Enhancing Graph Representation Learning with Localized Topological Features](https://mlsanity.com/paper/2501.09178/) (2025)
- [Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology](https://mlsanity.com/paper/2109.10960/) (2021)
- [k-Means Clustering for Persistent Homology](https://mlsanity.com/paper/2210.10003/) (2022)
- [Visualizing Topological Importance: A Class-Driven Approach](https://mlsanity.com/paper/2309.13185/) (2023)
- [Nonparametric regression on random geometric graphs sampled from submanifolds](https://mlsanity.com/paper/2405.20909/) (2024)
- [The Density of Cross-Persistence Diagrams and Its Applications](https://mlsanity.com/paper/2603.11623/) (2026)
- [Improved Modeling of Persistence Diagram](https://mlsanity.com/paper/2205.10907/) (2022)
- [Learning on Persistence Diagrams as Radon Measures](https://mlsanity.com/paper/2212.08295/) (2022)
- [Topological Analysis for Detecting Anomalies (TADA) in Time Series](https://mlsanity.com/paper/2406.06168/) (2024)
- [Revisiting the Sliced Wasserstein Kernel for persistence diagrams: a Figalli-Gigli approach](https://mlsanity.com/paper/2602.06539/) (2026)
- [Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams](https://mlsanity.com/paper/2607.27126/) (2026)

## Abstract (arXiv)

Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane ℝ2 that can equivalently be seen as discrete measures in ℝ2. When the data come as a random point cloud, these discrete measures become random measures whose expectation is studied in this paper. First, we show that for a wide class of filtrations, including the vCech and Rips-Vietoris filtrations, the expected persistence diagram, that is a deterministic measure on ℝ2 , has a density with respect to the Lebesgue measure. Second, building on the previous result we show that the persistence surface recently introduced in [Adams & al., Persistence images: a stable vector representation of persistent homology] can be seen as a kernel estimator of this density. We propose a cross-validation scheme for selecting an optimal bandwidth, which is proven to be a consistent procedure to estimate the density.