论文标题
从Varadhan的极限到本征图：多种学习背后的几何分析指南
From Varadhan's Limit to Eigenmaps: A Guide to the Geometric Analysis behind Manifold Learning
论文作者
论文摘要
我们概述了对黎曼流形的热内核和本征函数的历史，以及该理论如何通过特征图和其他光谱嵌入来分析高维数据的现代方法。我们从Varadhan定理开始，将热内核与Riemannian歧管上的距离函数有关。然后，我们回顾各种定理，这些定理绑定了riemannian歧管类别的热核。接下来，我们转向特征性函数，使用本征函数的热核的sturm-liouville分解以及各种定理，这些定理在riemannian歧管的类别上结合了本征函数。我们回顾了Riemannian流形的融合的各种概念，以及哪种Riemannian歧管相对于哪些收敛概念紧凑。然后，我们提出了贝拉德·贝森·盖洛特（Bérard-Besson-Gallot）的热核嵌入，嵌入了riemannian歧管和这些嵌入的截断。最后，我们将光谱嵌入的应用程序缩小到位于高维空间审查的数据集的尺寸，特别是Belkin-Niyogi和Coifman-Lafon的工作。我们还回顾了图形的光谱理论以及Dodziuk和Chung等的工作。我们与最新的Portegies定理和第一作者的定理结束，将截短的光谱嵌入在Riemannian歧管的关键类别上均匀。在整个过程中，我们提供了许多明确计算的示例和图形，并尝试提供尽可能完整的一组参考。我们希望纯净和应用的数学家都可以访问本文，从事几何分析及其博士学生。
We present an overview of the history of the heat kernel and eigenfunctions on Riemannian manifolds and how the theory has lead to modern methods of analyzing high dimensional data via eigenmaps and other spectral embeddings. We begin with Varadhan's Theorem relating the heat kernel to the distance function on a Riemannian manifold. We then review various theorems which bound the heat kernel on classes of Riemannian manifolds. Next we turn to eigenfunctions, the Sturm-Liouville Decomposition of the heat kernel using eigenfunctions, and various theorems which bound eigenfunctions on classes of Riemannian manifolds. We review various notions of convergence of Riemannian manifolds and which classes of Riemannian manifolds are compact with respect to which notions of convergence. We then present Bérard-Besson-Gallot's heat kernel embeddings of Riemannian manifolds and the truncation of those embeddings. Finally we turn to Applications of Spectral embeddings to the Dimension Reduction of data sets lying in high dimensional spaces reviewing, in particular, the work of Belkin-Niyogi and Coifman-Lafon. We also review the Spectral Theory of Graphs and the work of Dodziuk and Chung and others. We close with recent theorems of Portegies and of the first author controlling truncated spectral embeddings uniformly on key classes of Riemannian manifolds. Throughout we provide many explicitly computed examples and graphics and attempt to provide as complete a set of references as possible. We hope that this article is accessible to both pure and applied mathematicians working in Geometric Analysis and their doctoral students.
