How well behaved is finite dimensional Diffusion Maps?
This provides a theoretical foundation for the reliability of Diffusion Maps in practical applications, but it is incremental as it builds on existing methods with new error analyses.
The paper tackles the problem of quantifying geometric errors in finite-dimensional Diffusion Maps embeddings of submanifolds, establishing rigorous bounds on embedding errors as O((log n/n)^(1/(8d+16))) and on tangent space estimation errors with a similar rate.
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.