# How well behaved is finite dimensional Diffusion Maps?

arXiv: https://arxiv.org/abs/2412.03992 · HTML: https://mlsanity.com/paper/2412.03992/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 41 references that are in the map.
- First posted on arXiv 5 Dec 2024.

## Builds on (most cited first, 5 of 5)

- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Lipschitz regularity of graph Laplacians on random data clouds](https://mlsanity.com/paper/2007.06679/) (2020)
- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [The embedding dimension of Laplacian eigenfunction maps](https://mlsanity.com/paper/1605.01643/) (2016)
- [Spectral Convergence Rate of Graph Laplacian](https://mlsanity.com/paper/1510.08110/) (2015)

## Abstract (arXiv)

Under a set of assumptions on a family of submanifolds subset mathbb RD, we derive a series of geometric properties that remain valid after finite-dimensional and almost isometric Diffusion Maps (DM), including almost uniform density, finite polynomial approximation and reach. Leveraging these properties, we establish rigorous bounds on the embedding errors introduced by the DM algorithm is Oleft((fraclog nn)^frac18d+16right). Furthermore, we quantify the error between the estimated tangent spaces and the true tangent spaces over the submanifolds after the DM embedding, supP∈ PmathbbE_P^otimes tilden max_1≤ j angle (T_Yvarphi(M),jvarphi(M),hatTj)≤ tilden ≤ C left(fraclog n nright)^frack-1(8d+16)k, which providing a precise characterization of the geometric accuracy of the embeddings. These results offer a solid theoretical foundation for understanding the performance and reliability of DM in practical applications.