Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary

Andrew Lyons

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arXiv · 2501.09572

Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary

Andrew Lyons

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Abstract

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.

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Andrew Lyons. 2025-01-16. Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary. https://arxiv.org/abs/2501.09572

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