Convergence, optimization and stability of singular eigenmaps

TL;DR

This paper investigates the optimal scaling parameter for eigenmap approximation in nonlinear dimension reduction, using models and simulations to identify ranges that ensure accuracy across various geometric spaces.

Contribution

It provides an empirical approach to determine the approximately optimal epsilon for eigenmap approximation, supported by explicit models and Monte Carlo simulations.

Findings

Identifies epsilon ranges for accurate eigenmap approximation

Uses models like intervals, squares, tori, spheres, and fractals

Provides insights applicable to Riemannian manifolds and metric spaces

Abstract

Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $\epsilon$. If $\epsilon$ is too small or too large, then the approximation is inaccurate or completely breaks down. However, an analytic expression for the optimal $\epsilon$ is out of reach. In our work, we use some explicitly solvable models and Monte Carlo simulations to find the approximately optimal range of $\epsilon$ that gives, on average, relatively accurate approximation of the eigenmaps. Numerically we can consider several model situations where eigen-coordinates can be computed analytically, including intervals with uniform and weighted measures, squares, tori, spheres, and the Sierpinski gasket. In broader terms, we intend to study eigen-coordinates on weighted Riemannian manifolds, possibly with boundary, and on some metric measure spaces, such as fractals.