研究流形边界上局部线性嵌入的谱收敛，揭示其边界处的数学规律。
Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary
- 通过微分算子分析嵌入算法在边界区域的渐近行为
- 解析求解特征值极限并验证数值结果一致性
- 为其他紧流形提供可推广的变分框架，适合几何学习研究者
我们研究了当大规模数据从区间或圆盘采样时，无监督学习算法局部线性嵌入（Locally Linear Embedding）所对应的微分算子的特征值与特征函数。该微分算子为二阶、混合型且在边界附近退化。我们证明了特征函数的自然正则性条件会施加一致的边界条件，并利用Frobenius方法估计点态行为。随后，我们解析地确定了特征值的极限序列，并与数值预测进行比较。最后，我们提出一种适用于其他紧流形的变分框架以确定特征值。
原文摘要 · Abstract (English)
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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