标题： 有限维扩散映射有多良好？ 显示英文标题

标题： How well behaved is finite dimensional Diffusion Maps?

摘要： 在关于子流形族$\subset {\mathbb R}^D$的一组假设下，我们推导出一系列几何性质，在有限维和几乎等距扩散映射（DM）后仍然有效，包括几乎均匀密度、有限多项式逼近和可达性。 利用这些性质，我们建立了由 DM 算法引入的嵌入误差的严格界限，这是$O\left((\frac{\log n}{n})^{\frac{1}{8d+16}}\right)$。 此外，我们量化了 DM 嵌入后在子流形上估计切空间与真实切空间之间的误差， $\sup_{P\in \mathcal{P}}\mathbb{E}_{P^{\otimes \tilde{n}}} \max_{1\leq j \angle (T_{Y_{\varphi(M),j}}\varphi(M),\hat{T}_j)\leq \tilde{n}} \leq C \left(\frac{\log n }{n}\right)^\frac{k-1}{(8d+16)k}$， 这提供了对嵌入几何精度的精确描述。 这些结果为理解 DM 在实际应用中的性能和可靠性提供了坚实的理论基础。 显示英文摘要

摘要： 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.