证明了扩散映射在有限维下仍能保持几何精度，误差有严格数学界。
How well behaved is finite dimensional Diffusion Maps?
- 基于流形假设，推导出扩散映射的密度、逼近与曲率保持性质。
- 嵌入误差上限为O((log n/n)^{1/(8d+16)})，随样本数提升而下降。
- 首次精确量化切空间估计误差，适合理论研究与高维数据分析者。
在一组关于子流形的假设下，我们推导出有限维几乎等距扩散映射（DM）仍保持的几何性质，包括近似均匀密度、有限多项式逼近能力和可达性。利用这些性质，我们建立了扩散映射算法引入的嵌入误差的严格上界：O((log n/n)^{1/(8d+16)})。此外，我们量化了嵌入后估计切空间与真实切空间之间的误差：对所有P∈𝒫，E_{P^{⊗˜n}} max_{1≤j≤˜n} ∠(T_{Y_{φ(M),j}}φ(M), ˆT_j) ≤ C (log n/n)^{(k−1)/((8d+16)k)}，精确刻画了嵌入的几何保真度。这些结果为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。
原文摘要 · Abstract (English)
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, $\sup_{P\in \mathcal{P}}\mathbb{E}_{P^{\otimes \tilde{n}}} \max_{1\leq j \angle (T_{Y_{φ(M),j}}φ(M),\hat{T}_j)\leq \tilde{n}} \leq C \left(\frac{\log n }{n}\right)^\frac{k-1}{(8d+16)k}$, 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.
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