# The embedding dimension of Laplacian eigenfunction maps

arXiv: https://arxiv.org/abs/1605.01643 · HTML: https://mlsanity.com/paper/1605.01643/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 May 2016.

## Built on by (most cited first, 8 of 8)

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- [Convergence of Laplacian Eigenmaps and its Rate for Submanifolds with Singularities](https://mlsanity.com/paper/2110.08138/) (2021)
- [Sketching the Heat Kernel: Using Gaussian Processes to Embed Data](https://mlsanity.com/paper/2403.07929/) (2024)
- [Boundary Detection Algorithm Inspired by Locally Linear Embedding](https://mlsanity.com/paper/2406.18456/) (2024)
- [How well behaved is finite dimensional Diffusion Maps?](https://mlsanity.com/paper/2412.03992/) (2024)
- [Diffusion Maps is not Dimensionality Reduction](https://mlsanity.com/paper/2603.28037/) (2026)

## Abstract (arXiv)

Any closed, connected Riemannian manifold M can be smoothly embedded by its Laplacian eigenfunction maps into ℝm for some m. We call the smallest such m the maximal embedding dimension of M. We show that the maximal embedding dimension of M is bounded from above by a constant depending only on the dimension of M, a lower bound for injectivity radius, a lower bound for Ricci curvature, and a volume bound. We interpret this result for the case of surfaces isometrically immersed in ℝ3, showing that the maximal embedding dimension only depends on bounds for the Gaussian curvature, mean curvature, and surface area. Furthermore, we consider the relevance of these results for shape registration.