# Hodge Laplacians and Hodge Diffusion Maps

arXiv: https://arxiv.org/abs/2504.07910 · HTML: https://mlsanity.com/paper/2504.07910/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- First posted on arXiv 10 Apr 2025.

## Abstract (arXiv)

We introduce Hodge Diffusion Maps, a novel manifold learning algorithm designed to analyze and extract topological information from high-dimensional data-sets. This method approximates the exterior derivative acting on differential forms, thereby providing an approximation of the Hodge Laplacian operator. Hodge Diffusion Maps extend existing non-linear dimensionality reduction techniques, including vector diffusion maps, as well as the theories behind diffusion maps and Laplacian Eigenmaps. Our approach captures higher-order topological features of the data-set by projecting it into lower-dimensional Euclidean spaces using the Hodge Laplacian. We develop a theoretical framework to estimate the approximation error of the exterior derivative, based on sample points distributed over a real manifold. Numerical experiments support and validate the proposed methodology.