# A computationally efficient framework for vector representation of persistence diagrams

arXiv: https://arxiv.org/abs/2109.08239 · HTML: https://mlsanity.com/paper/2109.08239/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 16 Sep 2021.

## Builds on (most cited first, 9 of 9)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [Persistence Curves: A canonical framework for summarizing persistence diagrams](https://mlsanity.com/paper/1904.07768/) (2019)
- [A Nonparametric Approach for Multiple Change Point Analysis of Multivariate Data](https://mlsanity.com/paper/1306.4933/) (2013)
- [Functional Summaries of Persistence Diagrams](https://mlsanity.com/paper/1804.01618/) (2018)
- [ecp: An R Package for Nonparametric Multiple Change Point Analysis of Multivariate Data](https://mlsanity.com/paper/1309.3295/) (2013)
- [Topological Machine Learning with Persistence Indicator Functions](https://mlsanity.com/paper/1907.13496/) (2019)
- [Statistical Analysis of Persistence Intensity Functions](https://mlsanity.com/paper/1510.02502/) (2015)
- [Harnessing the power of Topological Data Analysis to detect change points in time series](https://mlsanity.com/paper/1910.12939/) (2019)

## Built on by (most cited first, 3 of 3)

- [Vector Summaries of Persistence Diagrams for Permutation-based Hypothesis Testing](https://mlsanity.com/paper/2306.06257/) (2023)
- [Vectorization of Persistence Diagrams for Topological Data Analysis in R and Python Using TDAvec Package](https://mlsanity.com/paper/2411.17340/) (2024)
- [Persistent Cross Entropy](https://mlsanity.com/paper/2608.24549/) (2026)

## Abstract (arXiv)

In Topological Data Analysis, a common way of quantifying the shape of data is to use a persistence diagram (PD). PDs are multisets of points in ℝ2 computed using tools of algebraic topology. However, this multi-set structure limits the utility of PDs in applications. Therefore, in recent years efforts have been directed towards extracting informative and efficient summaries from PDs to broaden the scope of their use for machine learning tasks. We propose a computationally efficient framework to convert a PD into a vector in ℝn, called a vectorized persistence block (VPB). We show that our representation possesses many of the desired properties of vector-based summaries such as stability with respect to input noise, low computational cost and flexibility. Through simulation studies, we demonstrate the effectiveness of VPBs in terms of performance and computational cost within various learning tasks, namely clustering, classification and change point detection.