# Vector Summaries of Persistence Diagrams for Permutation-based Hypothesis Testing

arXiv: https://arxiv.org/abs/2306.06257 · HTML: https://mlsanity.com/paper/2306.06257/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 45 references that are in the map.
- First posted on arXiv 9 Jun 2023.

## Builds on (most cited first, 17 of 17)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [Persistence Curves: A canonical framework for summarizing persistence diagrams](https://mlsanity.com/paper/1904.07768/) (2019)
- [Functional Summaries of Persistence Diagrams](https://mlsanity.com/paper/1804.01618/) (2018)
- [Embeddings of Persistence Diagrams into Hilbert Spaces](https://mlsanity.com/paper/1905.05604/) (2019)
- [Hypothesis Testing for Topological Data Analysis](https://mlsanity.com/paper/1310.7467/) (2013)
- [On the Metric Distortion of Embedding Persistence Diagrams into separable Hilbert spaces](https://mlsanity.com/paper/1806.06924/) (2018)
- [Statistical Analysis of Persistence Intensity Functions](https://mlsanity.com/paper/1510.02502/) (2015)
- [Approximation algorithms for 1-Wasserstein distance between persistence diagrams](https://mlsanity.com/paper/2104.07710/) (2021)
- [A computationally efficient framework for vector representation of persistence diagrams](https://mlsanity.com/paper/2109.08239/) (2021)
- [Introduction to the R package TDA](https://mlsanity.com/paper/1411.1830/) (2014)
- [Hypothesis Testing for Shapes using Vectorized Persistence Diagrams](https://mlsanity.com/paper/2006.05466/) (2020)
- [Functional central limit theorems for persistent Betti numbers on cylindrical networks](https://mlsanity.com/paper/2003.13490/) (2020)
- [A fast topological approach for predicting anomalies in time-varying graphs](https://mlsanity.com/paper/2305.06523/) (2023)

## Built on by (most cited first, 1 of 1)

- [From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Features](https://mlsanity.com/paper/2606.11911/) (2026)

## Abstract (arXiv)

Over the past decade, the techniques of topological data analysis (TDA) have grown into prominence to describe the shape of data. In recent years, there has been increasing interest in developing statistical methods and in particular hypothesis testing procedures for TDA. Under the statistical perspective, persistence diagrams -- the central multi-scale topological descriptors of data provided by TDA -- are viewed as random observations sampled from some population or process. In this context, one of the earliest works on hypothesis testing focuses on the two-group permutation-based approach where the associated loss function is defined in terms of within-group pairwise bottleneck or Wasserstein distances between persistence diagrams (Robinson and Turner, 2017). However, in situations where persistence diagrams are large in size and number, the permutation test in question gets computationally more costly to apply. To address this limitation, we instead consider pairwise distances between vectorized functional summaries of persistence diagrams for the loss function. In the present work, we explore the utility of the Betti function in this regard, which is one of the simplest function summaries of persistence diagrams. We introduce an alternative vectorization method for the Betti function based on integration and prove stability results with respect to the Wasserstein distance. Moreover, we propose a new shuffling technique of group labels to increase the power of the test. Through several experimental studies, on both synthetic and real data, we show that the vectorized Betti function leads to competitive results compared to the baseline method involving the Wasserstein distances for the permutation test.