Related papers: How well behaved is finite dimensional Diffusion Maps?

How well behaved is finite dimensional Diffusion Maps?

URL: http://arxiv.org/abs/2412.03992v1
Date: Thu, 05 Dec 2024 09:12:25 GMT
Title: How well behaved is finite dimensional Diffusion Maps?
Authors: Wenyu Bo, Marina Meilă,
Abstract summary: We derive a series of properties that remain valid after finite-dimensional and almost isometric Diffusion Maps (DM) We establish rigorous bounds on the embedding errors introduced by the DM algorithm is $Oleft(fraclog nn)frac18d+16right$. These results offer a solid theoretical foundation for understanding the performance and reliability of DM in practical applications.
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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 local reach. Leveraging these properties, we establish rigorous bounds on the embedding errors introduced by the DM algorithm is $O\left((\frac{\log n}{n})^{\frac{1}{8d+16}}\right)$. These results offer a solid theoretical foundation for understanding the performance and reliability of DM in practical applications.

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