Authors:Wenyu Bo, Marina Meilă (Department of Statistics University of Washington Seattle, WA)
Abstract:Under a set of assumptions on a family of submanifolds $\subset {\mathbb R}^D$, 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 $O\left((\frac{\log n}{n})^{\frac{1}{8d+16}}\right)$. Furthermore, we quantify the error between the estimated tangent spaces and the true tangent spaces over the submanifolds after the DM embedding,
$\sup_{P\in \mathcal{P}}\mathbb{E}_{P^{\otimes \tilde{n}}} \max_{1\leq j \angle (T_{Y_{\varphi(M),j}}\varphi(M),\hat{T}_j)\leq \tilde{n}} \leq C \left(\frac{\log n }{n}\right)^\frac{k-1}{(8d+16)k}$,
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.
| Comments: | 33 pages, 4 figures |
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST) |
| Cite as: | arXiv:2412.03992 [stat.ML] |
| (or arXiv:2412.03992v3 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2412.03992 arXiv-issued DOI via DataCite |
Submission history
From: Wenyu Bo [view email]
[v1]
Thu, 5 Dec 2024 09:12:25 UTC (32 KB)
[v2]
Fri, 21 Mar 2025 20:28:31 UTC (51 KB)
[v3]
Wed, 13 May 2026 18:46:35 UTC (47 KB)