Hodge Laplacians and Hodge Diffusion Maps
This work addresses the need for advanced topological analysis in data science, though it appears incremental as it extends existing techniques like diffusion maps.
The authors tackled the problem of extracting topological information from high-dimensional datasets by introducing Hodge Diffusion Maps, a manifold learning algorithm that approximates the Hodge Laplacian to capture higher-order features, with numerical experiments validating the method.
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.