How well behaved is finite dimensional Diffusion Maps?

2024-12-05

arXiv.org

AI Summary

This work investigates the geometric fidelity of finite-dimensional diffusion maps (DM) on submanifolds, focusing on quantitative analysis of embedding error and tangent space estimation error. Methodologically, it integrates tools from differential geometry, spectral graph theory, and probabilistic concentration inequalities. The main contributions are: (i) the first explicit convergence bound for DM embedding error—$Oig((log n/n)^{1/(8d+16)}ig)$—established under mild regularity assumptions; and (ii) a rigorous upper bound on tangent space estimation error. Results demonstrate that DM achieves approximate isometric embedding under appropriate sampling and parameter conditions, preserving key geometric properties including uniform density, polynomial approximation capability, and reach. The derived convergence rate is currently optimal for DM-based manifold learning. Collectively, this work provides the first theoretical framework for DM that simultaneously guarantees geometric accuracy and quantifiable statistical convergence rates, advancing foundational understanding of high-dimensional data dimensionality reduction and manifold modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Data Compression
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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 $Oleft((frac{log n}{n})^{frac{1}{8d+16}} ight)$. Furthermore, we quantify the error between the estimated tangent spaces and the true tangent spaces over the submanifolds after the DM embedding, $sup_{Pin mathcal{P}}mathbb{E}_{P^{otimes ilde{n}}} max_{1leq j angle (T_{Y_{varphi(M),j}}varphi(M),hat{T}_j)leq ilde{n}} leq C left(frac{log n }{n} ight)^frac{k-1}{(8d+16)k}$, 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.

Problem

• Analyzing finite-dimensional Diffusion Maps behavior
• Establishing bounds on DM embedding errors
• Quantifying tangent space estimation accuracy

Innovation

• Finite-dimensional Diffusion Maps geometric properties
• Embedding error bounds with DM algorithm
• Tangent space estimation error quantification