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.

翻译：拉普拉斯学习是一种半监督方法，通过利用未标记数据点提供的几何结构，从部分标记数据集中寻找缺失标签的解决方案。该方法最小化基于完整数据集构建的（离散）图上定义的狄利克雷能量。在有限维情形下，大数据（未标记）极限的渐近性质已得到充分理解，能够证明从图设置到由数据生成测度的勒贝格密度加权的连续索伯列夫半范数的收敛性。由于无限维空间缺乏勒贝格测度，当数据非有限维时需要重新思考分析方法。本文通过分析数据由希尔伯特空间上的高斯测度生成的情形，并证明图狄利克雷能量的逐点收敛性，朝着该方向迈出了第一步。