# A Notion of Harmonic Clustering in Simplicial Complexes

@article{Ebli2019ANO, title={A Notion of Harmonic Clustering in Simplicial Complexes}, author={Stefania Ebli and Gard Spreemann}, journal={2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA)}, year={2019}, pages={1083-1090} }

We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of, graph spectral clustering. The algorithm involves only sparse eigenproblems, and is therefore computationally efficient. We believe that it has broad application as a way to extract features from simplicial complexes that often arise in topological data analysis.

## 12 Citations

### Go with the Flow? A Large-Scale Analysis of Health Care Delivery Networks in the United States Using Hodge Theory

- Medicine2021 IEEE International Conference on Big Data (Big Data)
- 2021

A large-scale analysis of care delivery networks across the United States using the discrete Hodge decomposition, an emerging method of topological data analysis finds that the relative importance of each subspace is predictive of local care cost and quality, with outcomes tending to be better with greater curl flow and worse with greater harmonic flow.

### The decomposition of the higher-order homology embedding constructed from the k-Laplacian

- Computer Science, MathematicsNeurIPS
- 2021

An algorithm to factorize the homology embedding into subspaces corresponding to a manifold’s simplest topological components and is applied to the shortest homologous loop detection problem, a problem known to be NP-hard in general.

### Parallel decomposition of persistence modules through interval bases

- MathematicsArXiv
- 2021

An algorithm to decompose any finite-type persistence module with coefficients in a field into what is called an interval basis is introduced, which yields both the standard persistence pairs of Topological Data Analysis (TDA) and a special set of generators inducing the interval decomposition of the Structure theorem.

### Topological Point Cloud Clustering

- Computer ScienceArXiv
- 2023

Topological Point Cloud Clustering is presented, a new method to cluster points in an arbitrary point cloud based on their contribution to global topological features and is based on considering the spectral properties of a simplicial complex associated to the considered point cloud.

### Quantifying the structural stability of simplicial homology

- MathematicsArXiv
- 2023

. The homology groups of a simplicial complex reveal fundamental properties of the topology of the data or the system and the notion of topological stability naturally poses an important yet not…

### Harmonic representatives in homology over arbitrary fields

- MathematicsJournal of Applied and Computational Topology
- 2023

We introduce a notion of harmonic chain for chain complexes over fields of positive characteristic. A list of conditions for when a Hodge decomposition theorem holds in this setting is given and we…

### Link Partitioning on Simplicial Complexes Using Higher-Order Laplacians

- Computer Science2022 IEEE International Conference on Data Mining (ICDM)
- 2022

This paper introduces a link partitioning method that leverages higher-order (i.e. triadic and higher) information in simplicial complexes for better community detection and offers new theoretical results on the spectral properties of simplicial complex by studying the spectrum of the link random walk.

### Generalized Spectral Coarsening

- Mathematics, Computer ScienceArXiv
- 2022

A generalized spectral coarsening method that considers multiple Laplacian operators defined in different dimensionalities in tandem that allows controllable preservation of salient features from the high-resolution geometry and can therefore be customized to different applications.

### Dist2Cycle: A Simplicial Neural Network for Homology Localization

- Computer Science, MathematicsAAAI
- 2022

The proposed model enables learning topological features of the underlying simplicial complexes, specifically, the distance of each k-simplex from the nearest "optimal" k-th homology generator, effectively providing an alternative to homology localization.

### Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond

- Computer ScienceSignal Process.
- 2021

## 46 References

### Geometric Data Analysis Across Scales via Laplacian Eigenvector Cascading.

- Computer Science
- 2018

An algorithmic framework for constructing consistent multiscale Laplacian eigenfunctions (vectors) on data is developed, and it is shown via examples that cascading accelerates the computation of graph Laplacan eigenvectors, and more importantly, that one obtains consistent bases of the associated eigenspaces across scales.

### A tutorial on spectral clustering

- Computer ScienceStat. Comput.
- 2007

This tutorial describes different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches.

### Density-Connected Subspace Clustering for High-Dimensional Data

- Computer ScienceSDM
- 2004

SUBCLU (density-connected Subspace Clustering), an effective and efficient approach to the subspace clustering problem, based on a formal clustering notion using the concept of density-connectivity underlying the algorithm DBSCAN [EKSX96].

### Total variation and cheeger cuts

- Computer ScienceICML 2010
- 2010

This work gives a continuous relaxation of the Cheeger cut problem on a weighted graph and describes an algorithm for finding good cuts suggested by the similarities of the energy of the relaxed problem and various well studied energies in image processing.

### Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen

- Mathematics
- 1927