How well behaved is finite dimensional Diffusion Maps?

TL;DR

This paper provides a rigorous theoretical analysis of finite-dimensional Diffusion Maps, establishing bounds on embedding errors and tangent space estimation accuracy under certain geometric assumptions.

Contribution

It derives geometric properties preserved by Diffusion Maps and quantifies embedding and tangent space errors with explicit bounds, enhancing understanding of DM's reliability.

Findings

Embedding error bound: $O((rac{ ext{log } n}{n})^{rac{1}{8d+16}})$

Tangent space estimation error bound: $C (rac{ ext{log } n}{n})^{rac{k-1}{(8d+16)k}}$

Geometric properties like almost uniform density and reach are preserved after DM embedding.

Abstract

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.