# The mergegram of a dendrogram and its stability

arXiv: https://arxiv.org/abs/2007.11278 · HTML: https://mlsanity.com/paper/2007.11278/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jul 2020.

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

- [Recognizing Rigid Patterns of Unlabeled Point Clouds by Complete and Continuous Isometry Invariants with no False Negatives and no False Positives](https://mlsanity.com/paper/2303.15385/) (2023)
- [Beyond explaining: XAI-based Adaptive Learning with SHAP Clustering for Energy Consumption Prediction](https://mlsanity.com/paper/2402.04982/) (2024)
- [Quotient Geometry and Persistence-Stable Metrics for Swarm Configurations](https://mlsanity.com/paper/2603.18041/) (2026)

## Abstract (arXiv)

This paper extends the key concept of persistence within Topological Data Analysis (TDA) in a new direction. TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the 0-dimensional case the distance-based persistence is determined by a single-linkage (SL) clustering of a finite set in a metric space. Equivalently, the 0D persistence captures only edge-lengths of a Minimum Spanning Tree (MST). Both SL dendrogram and MST are unstable under perturbations of points. We define the new stable-under-noise mergegram, which outperforms previous isometry invariants on a classification of point clouds by PersLay.