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How well behaved is finite dimensional Diffusion Maps?

Wenyu Bo, Marina Meilă

TL;DR

This work analyzes finite-dimensional Diffusion Maps (DM) of a well-behaved family of manifolds embedded in Euclidean space and establishes that DM preserves key geometric properties such as near-uniform density, uniform polynomial reach, and controlled curvature. By leveraging these properties, the authors derive a finite-sample embedding error bound $O\left((\frac{\log n}{n})^{\frac{1}{8d+16}}\right)$ and a tangent-space estimation bound $\sup_{P\in \mathcal{P}} \mathbb{E}_{P^{\otimes \tilde{n}}} \max_{1\le j\le \tilde{n}} \angle\left(T_{Y_{\varphi(M),j}}\varphi(M),\hat{T}_j\right)\le C\left(\frac{\log n}{n}\right)^{\frac{k-1}{(8d+16)k}}$, where $\tilde{n}$ scales judiciously with $n$. The analysis ties together spectral convergence of DM, pushforward density regularity, and geometric quantities like reach and injectivity radius, culminating in a rigorous description of the geometric accuracy of DM embeddings. The results provide a theoretical foundation for reliable DM-based dimensionality reduction and tangent-space estimation in applications, with explicit rates depending on the manifold dimension $d$, smoothness $k$, and sample size $n$. Overall, the paper advances the understanding of DM stability in finite-sample regimes and informs practical choices of embedding dimension and sample allocation.

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.

How well behaved is finite dimensional Diffusion Maps?

TL;DR

This work analyzes finite-dimensional Diffusion Maps (DM) of a well-behaved family of manifolds embedded in Euclidean space and establishes that DM preserves key geometric properties such as near-uniform density, uniform polynomial reach, and controlled curvature. By leveraging these properties, the authors derive a finite-sample embedding error bound and a tangent-space estimation bound , where scales judiciously with . The analysis ties together spectral convergence of DM, pushforward density regularity, and geometric quantities like reach and injectivity radius, culminating in a rigorous description of the geometric accuracy of DM embeddings. The results provide a theoretical foundation for reliable DM-based dimensionality reduction and tangent-space estimation in applications, with explicit rates depending on the manifold dimension , smoothness , and sample size . Overall, the paper advances the understanding of DM stability in finite-sample regimes and informs practical choices of embedding dimension and sample allocation.

Abstract

Under a set of assumptions on a family of submanifolds , 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 . Furthermore, we quantify the error between the estimated tangent spaces and the true tangent spaces over the submanifolds after the DM embedding, , 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.

Paper Structure

This paper contains 33 sections, 19 theorems, 169 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background, challenges and assumptions
  3. Manifolds and the Diffusion Map
  4. The Diffusion map
  5. Assumptions
  6. Direct Consequences of Assumptions
  7. Flowchart
  8. Properties of the diffusion maps embeeding $\varphi(\mathcal{M})$
  9. Bounds on $\lambda_m$ and the (higher) derivatives of $\phi$
  10. Upper Bound of $\lambda_m$
  11. Estimate of Higher Order Derivatives
  12. Uniform upper bounds $\left\{L_{\perp},L_3,\dots,L_k\right\}$ on the higher derivatives of the local parametrization reminder
  13. Bounds on the pushforward density
  14. Estimation of $\tau_{min}$
  15. Local Reach
  16. ...and 18 more sections

Key Result

Theorem 1

Let $\mathcal{M}$ be the set of $d$ dimensional, closed Riemannian manifolds whose Ricci curvature is bounded from below by $\kappa$, injectivity radius is bounded from below by $\iota$, and the volume is bounded from above by $V$, we define $\varphi: M\subset \mathbb{R}^D \rightarrow N\subset \math then there exists a $t_0=t_0(d,\kappa,\iota,\epsilon)$ such that for all $0<t<t_0$, there exists a

Figures (4)

  • Figure 1: Flowchart
  • Figure 2: Isometry
  • Figure 3: Reach
  • Figure 4: Local Estimate

Theorems & Definitions (33)

  • Definition 1: Tangent Space
  • Definition 2: Geodesic Normal Coordinate
  • Definition 3: Riemannian Metric
  • Definition 4: Reachfederer1959curvature
  • Definition 5: Laplacian-Beltrami Operator
  • Theorem 1
  • Lemma 2
  • Corollary 3: Complete Manifold
  • Corollary 4: The injectivity radius
  • Corollary 5: Diameter
  • ...and 23 more