laplacian-eigenmaps-laplace-beltrami-convergence
IN premise — summaries/2026/08/24/wiki-Nonlinear_dimensionality_reduction-chunk-1.md
Created 2026-08-24T17:11:20+00:00
Laplacian Eigenmaps uses eigenfunctions of the Laplace–Beltrami operator as embedding dimensions, and the graph Laplacian converges to the Laplace–Beltrami operator as the number of samples N approaches infinity under mild conditions.
Summary
Laplacian Eigenmaps is not just a clever graph trick on finite data; as the dataset grows, the math it performs provably approaches the true geometric structure of the smooth surface the data came from. This means the algorithm's output is grounded in the actual shape of the underlying space, not an artifact of how the points happen to be sampled.