# Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary

arXiv: https://arxiv.org/abs/2501.09572 · HTML: https://mlsanity.com/paper/2501.09572/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- First posted on arXiv 16 Jan 2025.

## Abstract (arXiv)

We study the eigenvalues and eigenfunctions of a differential operator that governs the asymptotic behavior of the unsupervised learning algorithm known as Locally Linear Embedding when a large data set is sampled from an interval or disc. In particular, the differential operator is of second order, mixed-type, and degenerates near the boundary. We show that a natural regularity condition on the eigenfunctions imposes a consistent boundary condition and use the Frobenius method to estimate pointwise behavior. We then determine the limiting sequence of eigenvalues analytically and compare them to numerical predictions. Finally, we propose a variational framework for determining eigenvalues on other compact manifolds.