How well behaved is finite dimensional Diffusion Maps?

Under a set of assumptions on a family of submanifolds subset mathbb RD, 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 Oleft((fraclog nn)^frac18d+16right). Furthermore, we quantify the error between the estimated tangent spaces and the true tangent spaces over the submanifolds after the DM embedding, supP∈ PmathbbE_P^otimes tilden max_1≤ j angle (T_Yvarphi(M),jvarphi(M),hatTj)≤ tilden ≤ C left(fraclog n nright)^frack-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.