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.

翻译：在一族子流形 $\subset {\mathbb R}^D$ 满足一组假设的前提下，我们推导出一系列在有限维且近似等距的扩散映射（DM）后仍保持有效的几何性质，包括近似均匀密度、有限多项式逼近与局部抵达距离。利用这些性质，我们严格建立了扩散映射算法引入的嵌入误差界为 $O\left((\frac{\log n}{n})^{\frac{1}{8d+16}}\right)$。这些结果为理解扩散映射在实际应用中的性能与可靠性提供了坚实的理论基础。