# Euler Characteristic Curves and Profiles: a stable shape invariant for big data problems

arXiv: https://arxiv.org/abs/2212.01666 · HTML: https://mlsanity.com/paper/2212.01666/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 41 references that are in the map.
- First posted on arXiv 3 Dec 2022.

## Builds on (most cited first, 10 of 10)

- [Persistence Curves: A canonical framework for summarizing persistence diagrams](https://mlsanity.com/paper/1904.07768/) (2019)
- [Persistence paths and signature features in topological data analysis](https://mlsanity.com/paper/1806.00381/) (2018)
- [An Introduction to Multiparameter Persistence](https://mlsanity.com/paper/2203.14289/) (2022)
- [The Simplex Tree: an Efficient Data Structure for General Simplicial Complexes](https://mlsanity.com/paper/2001.02581/) (2020)
- [Distributed computation of persistent homology](https://mlsanity.com/paper/1310.0710/) (2013)
- [Euler Characteristic Surfaces](https://mlsanity.com/paper/2102.08260/) (2021)
- [On the Bootstrap for Persistence Diagrams and Landscapes](https://mlsanity.com/paper/1311.0376/) (2013)
- [Instability of the Betti Sequence for Persistent Homology and a Stabilized Version of the Betti Sequence](https://mlsanity.com/paper/2109.09218/) (2021)
- [Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images](https://mlsanity.com/paper/1705.02045/) (2017)
- [GPU Computation of the Euler Characteristic Curve for Imaging Data](https://mlsanity.com/paper/2203.09087/) (2022)

## Built on by (most cited first, 8 of 8)

- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [TopER: Topological Embeddings in Graph Representation Learning](https://mlsanity.com/paper/2410.01778/) (2024)
- [CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizations](https://mlsanity.com/paper/2510.12795/) (2025)
- [T3former: Temporal Graph Classification with Topological Machine Learning](https://mlsanity.com/paper/2510.13789/) (2025)
- [Certified Topological Interaction in Neural Representations: Exact Tests and the Statistic They Require](https://mlsanity.com/paper/2609.08561/) (2026)
- [The Morse Transform for Discrete Shape Analysis](https://mlsanity.com/paper/2503.04507/) (2025)
- [CINNAMON: A hybrid approach to change point detection and parameter estimation in single-particle tracking data](https://mlsanity.com/paper/2503.14253/) (2025)
- [The Null Is the Hard Part: Exact Tests for Memorization in Generative Models](https://mlsanity.com/paper/2610.00251/) (2026)

## Abstract (arXiv)

Tools of Topological Data Analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well studied data summary, suffers a number of limitations; its computations are hard to distribute, it is hard to generalize to multifiltrations and is computationally prohibitive for big data-sets. In this paper we study the concept of Euler Characteristics Curves, for one parameter filtrations and Euler Characteristic Profiles, for multi-parameter filtrations. While being a weaker invariant in one dimension, we show that Euler Characteristic based approaches do not possess some handicaps of persistent homology; we show efficient algorithms to compute them in a distributed way, their generalization to multifiltrations and practical applicability for big data problems. In addition we show that the Euler Curves and Profiles enjoys certain type of stability which makes them robust tool in data analysis. Lastly, to show their practical applicability, multiple use-cases are considered.