# Helmholtzian Eigenmap: Topological feature discovery & edge flow learning from point cloud data

arXiv: https://arxiv.org/abs/2103.07626 · HTML: https://mlsanity.com/paper/2103.07626/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 48 references that are in the map.
- First posted on arXiv 13 Mar 2021.

## Builds on (most cited first, 11 of 11)

- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space](https://mlsanity.com/paper/1808.02111/) (2018)
- [Graph-based Semi-Supervised & Active Learning for Edge Flows](https://mlsanity.com/paper/1905.07451/) (2019)
- [Two's company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data](https://mlsanity.com/paper/1601.01704/) (2016)
- [Improved graph Laplacian via geometric self-consistency](https://mlsanity.com/paper/1406.0118/) (2014)
- [Graph Induced Complex on Point Data](https://mlsanity.com/paper/1304.0662/) (2013)
- [Approximate Cech Complexes in Low and High Dimensions](https://mlsanity.com/paper/1307.3272/) (2013)

## Built on by (most cited first, 7 of 7)

- [Deciphering Spatio-Temporal Graph Forecasting: A Causal Lens and Treatment](https://mlsanity.com/paper/2309.13378/) (2023)
- [Hodge-Compositional Edge Gaussian Processes](https://mlsanity.com/paper/2310.19450/) (2023)
- [The decomposition of the higher-order homology embedding constructed from the k-Laplacian](https://mlsanity.com/paper/2107.10970/) (2021)
- [Point-Level Topological Representation Learning on Point Clouds](https://mlsanity.com/paper/2406.02300/) (2024)
- [Topological Schr"odinger Bridge Matching](https://mlsanity.com/paper/2504.04799/) (2025)
- [Multiscale Hodge Scattering Networks for Data Analysis](https://mlsanity.com/paper/2311.10270/) (2023)
- [Topological Signal Processing: An Application-Oriented Tutorial](https://mlsanity.com/paper/2605.22853/) (2026)

## Abstract (arXiv)

The manifold Helmholtzian (1-Laplacian) operator Delta1 elegantly generalizes the Laplace-Beltrami operator to vector fields on a manifold mathcal M. In this work, we propose the estimation of the manifold Helmholtzian from point cloud data by a weighted 1-Laplacian mathcal L1. While higher order Laplacians have been introduced and studied, this work is the first to present a graph Helmholtzian constructed from a simplicial complex as a consistent estimator for the continuous operator in a non-parametric setting. Equipped with the geometric and topological information about mathcal M, the Helmholtzian is a useful tool for the analysis of flows and vector fields on mathcal M via the Helmholtz-Hodge theorem. In addition, the mathcal L1 allows the smoothing, prediction, and feature extraction of the flows. We demonstrate these possibilities on substantial sets of synthetic and real point cloud datasets with non-trivial topological structures; and provide theoretical results on the limit of mathcal L1 to Delta1.