# The persistence landscape and some of its properties

arXiv: https://arxiv.org/abs/1810.04963 · HTML: https://mlsanity.com/paper/1810.04963/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2018.

## Built on by (most cited first, 19 of 19)

- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [From Geometry to Topology: Inverse Theorems for Distributed Persistence](https://mlsanity.com/paper/2101.12288/) (2021)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Brain-Inspired Spiking Neural Network for Online Unsupervised Time Series Prediction](https://mlsanity.com/paper/2304.04697/) (2023)
- [TopoCL: Topological Contrastive Learning for Time Series](https://mlsanity.com/paper/2502.02924/) (2025)
- [The Shape of Adversarial Influence: Characterizing LLM Latent Spaces with Persistent Homology](https://mlsanity.com/paper/2505.20435/) (2025)
- [Classification of Stochastic Processes with Topological Data Analysis](https://mlsanity.com/paper/2206.03973/) (2022)
- [Time-Series Forecasting via Topological Information Supervised Framework with Efficient Topological Feature Learning](https://mlsanity.com/paper/2503.23757/) (2025)
- [Portfolio Selection via Topological Data Analysis](https://mlsanity.com/paper/2308.07944/) (2023)
- [Topological data analysis of human vowels: Persistent homologies across representation spaces](https://mlsanity.com/paper/2310.06508/) (2023)
- [Novel definition and quantitative analysis of branch structure with topological data analysis](https://mlsanity.com/paper/2402.07436/) (2024)
- [Invariant Representations of Embedded Simplicial Complexes](https://mlsanity.com/paper/2302.13565/) (2023)
- [Hierarchical Clustering in LambdaCDM Cosmologies via Persistence Energy](https://mlsanity.com/paper/2401.01988/) (2024)
- [Sparse Portfolio Selection via Topological Data Analysis based Clustering](https://mlsanity.com/paper/2401.16920/) (2024)
- [Steering Optimisation Trajectories in Diffusion Representation Learning](https://mlsanity.com/paper/2607.05319/) (2026)
- [Denoising 3D images: robustness of persistent homology measures](https://mlsanity.com/paper/2607.24579/) (2026)
- [Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams](https://mlsanity.com/paper/2607.27126/) (2026)
- [Data-driven techniques for translational neuroscience and personalized neuro-health](https://mlsanity.com/paper/2608.13749/) (2026)
- [PIT-GCL: Protein Interaction using Topological Graph Contrastive Learning](https://mlsanity.com/paper/2610.04850/) (2026)

## Abstract (arXiv)

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.