We investigate a family of regression problems in a semi-supervised setting. The task is to assign real-valued labels to a set of $n$ sample points, provided a small training subset of $N$ labeled points. A goal of semi-supervised learning is to take advantage of the (geometric) structure provided by the large number of unlabeled data when assigning labels. We consider a random geometric graph, with connection radius $\epsilon(n)$, to represent the geometry of the data set. We study objective functions which reward the regularity of the estimator function and impose or reward the agreement with the training data. In particular we consider discrete $p$-Laplacian regularization. We investigate asymptotic behavior in the limit where the number of unlabeled points increases while the number of training points remains fixed. We uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. We rigorously obtain almost optimal ranges on the scaling of $\epsilon(n)$ for the asymptotic consistency to hold. We discover that for standard approaches used thus far there is a restrictive upper bound on how quickly $\epsilon(n)$ must converge to zero as $n \to \infty$. Furthermore we introduce a new model which overcomes this restriction. It is as simple as the standard models, but converges as soon as $\epsilon(n) \to 0$ as $n \to \infty$.

本研究探讨了半监督学习中的回归问题，以随机几何图形模拟数据几何结构，将离散的$p$-拉普拉斯正则化纳入模型，研究了无标记点数增加时渐近表现的性质，发现模型存在收敛性限制，提出了一个简单的模型来解决这一限制。

半监督学习中$p$-Laplacian正则化的分析