# Stochastic Convergence of Persistence Landscapes and Silhouettes

arXiv: https://arxiv.org/abs/1312.0308 · HTML: https://mlsanity.com/paper/1312.0308/

## Key facts

- Cited by 43 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2013.

## Built on by (most cited first, 30 of 43)

- [RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds](https://mlsanity.com/paper/2202.01725/) (2022)
- [TopP&R: Robust Support Estimation Approach for Evaluating Fidelity and Diversity in Generative Models](https://mlsanity.com/paper/2306.08013/) (2023)
- [ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery](https://mlsanity.com/paper/2211.03808/) (2022)
- [Persistent Topological Features in Large Language Models](https://mlsanity.com/paper/2410.11042/) (2024)
- [PHG-Net: Persistent Homology Guided Medical Image Classification](https://mlsanity.com/paper/2311.17243/) (2023)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Learning Hyperbolic Representations of Topological Features](https://mlsanity.com/paper/2103.09273/) (2021)
- [A Random Persistence Diagram Generator](https://mlsanity.com/paper/2104.07737/) (2021)
- [Improving Self-supervised Molecular Representation Learning using Persistent Homology](https://mlsanity.com/paper/2311.17327/) (2023)
- [Diffeomorphic interpolation for efficient persistence-based topological optimization](https://mlsanity.com/paper/2405.18820/) (2024)
- [Mapping the Multiverse of Latent Representations](https://mlsanity.com/paper/2402.01514/) (2024)
- [Rethinking Persistent Homology for Visual Recognition](https://mlsanity.com/paper/2207.04220/) (2022)
- [EMP: Effective Multidimensional Persistence for Graph Representation Learning](https://mlsanity.com/paper/2401.13713/) (2024)
- [Topological Causal Effects](https://mlsanity.com/paper/2603.02289/) (2026)
- [Topological EEG Nonlinear Dynamics Analysis for Emotion Recognition](https://mlsanity.com/paper/2203.06895/) (2022)
- [Do Neural Networks Trained with Topological Features Learn Different Internal Representations?](https://mlsanity.com/paper/2211.07697/) (2022)
- [Image Classification using Combination of Topological Features and Neural Networks](https://mlsanity.com/paper/2311.06375/) (2023)
- [Time-Aware Knowledge Representations of Dynamic Objects with Multidimensional Persistence](https://mlsanity.com/paper/2401.13157/) (2024)
- [Homological Time Series Analysis of Sensor Signals from Power Plants](https://mlsanity.com/paper/2106.02493/) (2021)
- [A topological classifier to characterize brain states: When shape matters more than variance](https://mlsanity.com/paper/2303.04231/) (2023)
- [Stability for Inference with Persistent Homology Rank Functions](https://mlsanity.com/paper/2307.02904/) (2023)
- [A Topological Machine Learning Pipeline for Classification](https://mlsanity.com/paper/2309.15276/) (2023)
- [Topological data analysis of human vowels: Persistent homologies across representation spaces](https://mlsanity.com/paper/2310.06508/) (2023)
- [Explaining the Power of Topological Data Analysis in Graph Machine Learning](https://mlsanity.com/paper/2401.04250/) (2024)
- [CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizations](https://mlsanity.com/paper/2510.12795/) (2025)
- [Topological descriptors of foot clearance gait dynamics improve differential diagnosis of Parkinsonism](https://mlsanity.com/paper/2603.06212/) (2026)
- [The Density of Cross-Persistence Diagrams and Its Applications](https://mlsanity.com/paper/2603.11623/) (2026)
- [Fixation Sequences as Time Series: A Topological Approach to Dyslexia Detection](https://mlsanity.com/paper/2604.21698/) (2026)
- [A Mathematical Framework for Topological Causal Data Analysis](https://mlsanity.com/paper/2607.28161/) (2026)
- [TopoFormer: Topology Meets Attention for Graph Learning](https://mlsanity.com/paper/2607.28259/) (2026)

## Abstract (arXiv)

Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we can summarize the persistent homology with the persistence landscape, introduced by Bubenik, which converts a diagram into a well-behaved real-valued function. We investigate the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence of the bootstrap. In addition, we introduce an alternate functional summary of persistent homology, which we call the silhouette, and derive an analogous statistical theory.