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Hodge Laplacians and Hodge Diffusion Maps by Gomez, Alvaro Almeida; Franco, Jorge Duque is a scholarly article available to read on EtoBox.

What is Hodge Laplacians and Hodge Diffusion Maps about?

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.

Author

Gomez, Alvaro Almeida; Franco, Jorge Duque

Published

2025

Language

EN