研究无限维高斯数据下拉普拉斯学习的极限行为，揭示其点态收敛性。
Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensions
- 基于图上的狄利克雷能量，分析高维高斯数据的半监督学习
- 证明了在无限维希尔伯特空间中图能量的点态收敛
- 为高维数据建模提供理论支撑，适合关注概率几何与机器学习交叉的研究者
拉普拉斯学习是一种半监督方法，通过利用未标记数据点的几何结构来推断部分标记数据集中的缺失标签。该方法最小化由完整数据集构建的（离散）图上的狄利克雷能量。在有限维情形下，当未标记数据量趋于无穷时，图上的能量收敛到由数据生成测度的勒贝格密度加权的连续型索博列夫半范数。由于无限维空间中缺乏勒贝格测度，若数据非有限维，则需重新思考分析框架。本文首次在数据由希尔伯特空间上的高斯测度生成的设定下展开分析，证明了图狄利克雷能量的逐点收敛性。
原文摘要 · Abstract (English)
Laplace learning is a semi-supervised method, a solution for finding missing labels from a partially labeled dataset utilizing the geometry given by the unlabeled data points. The method minimizes a Dirichlet energy defined on a (discrete) graph constructed from the full dataset. In finite dimensions the asymptotics in the large (unlabeled) data limit are well understood with convergence from the graph setting to a continuum Sobolev semi-norm weighted by the Lebesgue density of the data-generating measure. The lack of the Lebesgue measure on infinite-dimensional spaces requires rethinking the analysis if the data aren't finite-dimensional. In this paper we make a first step in this direction by analyzing the setting when the data are generated by a Gaussian measure on a Hilbert space and proving pointwise convergence of the graph Dirichlet energy.
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