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Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary

Andrew Lyons

TL;DR

The paper analyzes the spectral behavior of the LLE-approximating operator $D_\epsilon$ on manifolds with boundary, showing it is a second-order, mixed-type operator that degenerates at the boundary. Using Frobenius boundary-layer analysis, it derives a natural regularity condition that enforces a quasi-Neumann boundary condition on the elliptic region and yields analytic, convergent eigenvalues that mirror the LLE matrix for large data. The interval and disc are treated in parallel via Sturm–Liouville and separable-variables reductions, producing explicit eigenvalue conditions and a variational (min-max) interpretation. Numerical experiments corroborate the theoretical predictions, and the paper extends the framework to a general variational setting on compact manifolds with boundary, linking graph-based methods to classical spectral theory. This work clarifies how boundary effects influence spectral convergence in LLE and provides a robust framework for computing eigenvalues on other manifolds.

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.

Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary

TL;DR

The paper analyzes the spectral behavior of the LLE-approximating operator on manifolds with boundary, showing it is a second-order, mixed-type operator that degenerates at the boundary. Using Frobenius boundary-layer analysis, it derives a natural regularity condition that enforces a quasi-Neumann boundary condition on the elliptic region and yields analytic, convergent eigenvalues that mirror the LLE matrix for large data. The interval and disc are treated in parallel via Sturm–Liouville and separable-variables reductions, producing explicit eigenvalue conditions and a variational (min-max) interpretation. Numerical experiments corroborate the theoretical predictions, and the paper extends the framework to a general variational setting on compact manifolds with boundary, linking graph-based methods to classical spectral theory. This work clarifies how boundary effects influence spectral convergence in LLE and provides a robust framework for computing eigenvalues on other manifolds.

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.
Paper Structure (14 sections, 15 theorems, 105 equations, 7 figures)

This paper contains 14 sections, 15 theorems, 105 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Eigenfunction properties on the interval and disc.
  3. Outline.
  4. Acknowledgments.
  5. Construction and Properties of the LLE Matrix.
  6. LLE on the Interval
  7. Presentation of the differential operator on the interval.
  8. Proof of Theorem \ref{['thm:1d']}.
  9. LLE on the Disc
  10. Presentation of the differential operator on the disc.
  11. Proof of Theorem \ref{['thm:2d']}.
  12. Numerical Comparison
  13. Proof of Theorem \ref{['thm:blah']}.
  14. A Variational Convergence Example

Key Result

Theorem 1.1

Let $M$ be the unit interval $[0,1]$ and $M_+=\{x: p_1(x)>0\}$ with $p_1,w_1$ given in (eq:1). There exists a constant $\epsilon_0>0$ such that for all $0<\epsilon\leq \epsilon_0$, the set of $C^2(M)$ solutions to (eq:PDE) coincides with eigenfunctions of $D_\epsilon$ satisfying that for every $y\in If (eq:quasiInt) holds,

Figures (7)

  • Figure 1: When $M$ is the disc, the left figure depicts the boundary layer $M_\epsilon$, and the right figure depicts the elliptic and hyperbolic regions $M_+,M_-$ separated by $\Gamma$.
  • Figure 2: For $\epsilon=0.05$, the first $5$ eigenvectors of $I-W$ sampled on $20,000$ points in the interval are plotted above. Each is labeled with the respective eigenvalue, appropriately scaled by $\epsilon^{-2}$ according to (\ref{['eq:Pconv']}).
  • Figure 3: In the left figure, the first $8$ eigenvalues of $I-W$ are plotted for different data set sizes and fixed $\epsilon=0.05$. We consider eigenvalue sequences corresponding to $5,000$ points, $10,000$ points, and $20,000$ points before plotting the analytic expectation to display convergence. In the right figure, we plot the relative errors $\left|\lambda_j(I-W)-\lambda_j(D_\epsilon)\right|\left|\lambda_j(D_\epsilon)\right|^{-3/2}$ for each eigenvalue sequence.
  • Figure 4: For $\epsilon=0.05$, several eigenfunctions of $D_\epsilon$ with quasi-Neumann boundary conditions (\ref{['eq:quasiInt']}) plotted above. Each eigenfunction has been discretized over $20,000$ points and is normalized to minimize error with the corresponding eigenvector of $I-W$, featured in Figure \ref{['fig:IntLLE']}. From top left to bottom right, we display the $2$nd, $3$rd, $4$th, and $5$th eigenfunctions, each labeled with the sup error (when compared with the corresponding eigenvector of $I-W$) and analytically-predicted eigenvalue
  • Figure 5: For $\epsilon=0.05$, the first $48$ eigenvectors of $I-W$ sampled on $160,000$ points in the disc are plotted from top left to bottom right. Each is labeled with the respective eigenvalue, appropriately scaled by $\epsilon^{-2}$ according to (\ref{['eq:Pconv']}).
  • ...and 2 more figures

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Conjecture 1
  • Proposition 3.1
  • Proposition 3.2
  • proof : Proof of Proposition \ref{['lem:0e']}.
  • ...and 16 more