# Moving Up the Cluster Tree with the Gradient Flow

arXiv: https://arxiv.org/abs/2109.08362 · HTML: https://mlsanity.com/paper/2109.08362/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 73 references that are in the map.
- First posted on arXiv 17 Sep 2021.

## Builds on (most cited first, 8 of 8)

- [A Population Background for Nonparametric Density-Based Clustering](https://mlsanity.com/paper/1408.1381/) (2014)
- [Density Level Sets: Asymptotics, Inference, and Visualization](https://mlsanity.com/paper/1504.05438/) (2015)
- [Consistent procedures for cluster tree estimation and pruning](https://mlsanity.com/paper/1406.1546/) (2014)
- [The modal age of Statistics](https://mlsanity.com/paper/1807.02789/) (2018)
- [Nonparametric Inference For Density Modes](https://mlsanity.com/paper/1312.7567/) (2013)
- [Beyond Hartigan Consistency: Merge Distortion Metric for Hierarchical Clustering](https://mlsanity.com/paper/1506.06422/) (2015)
- [Fully adaptive density-based clustering](https://mlsanity.com/paper/1409.8437/) (2014)
- [Statistical Inference using the Morse-Smale Complex](https://mlsanity.com/paper/1506.08826/) (2015)

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

- [An Axiomatic Definition of Hierarchical Clustering](https://mlsanity.com/paper/2407.03574/) (2024)
- [Graph Max Shift: A Hill-Climbing Method for Graph Clustering](https://mlsanity.com/paper/2411.18794/) (2024)
- [Stable Density Ridges: Consistency and Convergence of Subspace Constrained Mean Shift](https://mlsanity.com/paper/2608.05112/) (2026)

## Abstract (arXiv)

The paper establishes a strong correspondence between two important clustering approaches that emerged in the 1970's: clustering by level sets or cluster tree as proposed by Hartigan and clustering by gradient lines or gradient flow as proposed by Fukunaga and Hostetler. We do so by showing that we can move up the cluster tree by following the gradient ascent flow.