# Cycles Communities from the Perspective of Dendrograms and Gradient Sampling

arXiv: https://arxiv.org/abs/2512.12974 · HTML: https://mlsanity.com/paper/2512.12974/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 36 references that are in the map.
- First posted on arXiv 15 Dec 2025.

## Builds on (most cited first, 12 of 12)

- [Hierarchical Clustering: Objective Functions and Algorithms](https://mlsanity.com/paper/1704.02147/) (2017)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [Volume Optimal Cycle: Tightest representative cycle of a generator on persistent homology](https://mlsanity.com/paper/1712.05103/) (2017)
- [Cycle Registration in Persistent Homology with Applications in Topological Bootstrap](https://mlsanity.com/paper/2101.00698/) (2021)
- [A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis](https://mlsanity.com/paper/2109.00530/) (2021)
- [Persistent Homology in Sparse Regression and its Application to Brain Morphometry](https://mlsanity.com/paper/1409.0177/) (2014)
- [Fast Topological Signal Identification and Persistent Cohomological Cycle Matching](https://mlsanity.com/paper/2209.15446/) (2022)
- [A Subsequence Approach to Topological Data Analysis for Irregularly-Spaced Time Series](https://mlsanity.com/paper/2410.13723/) (2024)
- [MaxTDA: Robust Statistical Inference for Maximal Persistence in Topological Data Analysis](https://mlsanity.com/paper/2504.03897/) (2025)
- [Discrete Heat Kernels on Simplicial Complexes and Its Application to Functional Brain Networks](https://mlsanity.com/paper/2509.16908/) (2025)
- [Topologically Invariant Permutation Test](https://mlsanity.com/paper/2511.06153/) (2025)

## Abstract (arXiv)

Identifying and comparing topological features, particularly cycles, across different topological objects remains a fundamental challenge in persistent homology and topological data analysis. This work introduces a novel framework for constructing cycle communities through two complementary approaches. First, a dendrogram-based methodology leverages merge-tree algorithms to construct hierarchical representations of homology classes from persistence intervals. The Wasserstein distance on merge trees is introduced as a metric for comparing dendrograms, establishing connections to hierarchical clustering frameworks. Through simulation studies, the discriminative power of dendrogram representations for identifying cycle communities is demonstrated. Second, an extension of Stratified Gradient Sampling simultaneously learns multiple filter functions that yield cycle barycenter functions capable of faithfully reconstructing distinct sets of cycles. The set of cycles each filter function can reconstruct constitutes cycle communities that are non-overlapping and partition the space of all cycles. Together, these approaches transform the problem of cycle matching into both a hierarchical clustering and topological optimization framework, providing principled methods to identify similar topological structures both within and across groups of topological objects.