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.

翻译：本文研究了一个微分算子的特征值与特征函数，该算子决定了当大规模数据集从区间或圆盘采样时，无监督学习算法局部线性嵌入的渐近行为。特别地，该微分算子为二阶混合型，且在边界附近退化。我们证明，特征函数的一个自然正则性条件会施加一致的边界条件，并利用弗罗贝尼乌斯方法估计其逐点行为。随后，我们解析地确定了特征值的极限序列，并将其与数值预测进行比较。最后，我们提出了一个变分框架，用于确定其他紧致流形上的特征值。