This paper addresses theory and applications of ${\ell }_p$-based Laplacian regularization in semi-supervised learning. The graph p-Laplacian for $p>2$ has been proposed recently as a replacement for the standard ($p=2$) graph Laplacian in semi-supervised learning problems with very few labels, where Laplacian learning is degenerate.

In the first part of the paper we prove new discrete to continuum convergence results for p-Laplace problems on k-nearest neighbor (k-NN) graphs, which are more commonly used in practice than random geometric graphs. Our analysis shows that, on k-NN graphs, the p-Laplacian retains information about the data distribution as $p\to \infty$ and Lipschitz learning ($p=\infty$) is sensitive to the data distribution. This situation can be contrasted with random geometric graphs, where the p-Laplacian forgets the data distribution as $p\to \infty$. We also present a general framework for proving discrete to continuum convergence results in graph-based learning that only requires pointwise consistency and monotonicity.

In the second part of the paper, we develop fast algorithms for solving the variational and game-theoretic p-Laplace equations on weighted graphs for $p>2$. We present several efficient and scalable algorithms for both formulations, and present numerical results on synthetic data indicating their convergence properties. Finally, we conduct extensive numerical experiments on the MNIST, FashionMNIST and EMNIST datasets that illustrate the effectiveness of the p-Laplacian formulation for semi-supervised learning with few labels. In particular, we find that Lipschitz learning ($p=\infty$) performs well with very few labels on k-NN graphs, which experimentally validates our theoretical findings that Lipschitz learning retains information about the data distribution (the unlabeled data) on k-NN graphs.

本文阐述了理论和应用${\ell }_p$半监督学习中的基于拉普拉斯正则化。图p -拉普拉斯算子为$p>2$最近被提议作为标准的替代品（$p=2$) 在标签很少的半监督学习问题中绘制拉普拉斯算子图，其中拉普拉斯算子学习是退化的。

在论文的第一部分，我们证明了 k-最近邻 (k-NN) 图上 p-Laplace 问题的新离散到连续收敛结果，这在实践中比随机几何图更常用。我们的分析表明，在k -NN 图上，p -拉普拉斯算子保留了有关数据分布的信息$p\to \infty$和 Lipschitz 学习（$p=\infty$) 对数据分布很敏感。这种情况可以与随机几何图形成对比，其中p-拉普拉斯算子忘记了数据分布$p\to \infty$. 我们还提出了一个通用框架，用于证明仅需要逐点一致性和单调性的基于图的学​​习中的离散到连续收敛结果。

在本文的第二部分，我们开发了快速算法来求解加权图上的变分和博弈p-拉普拉斯方程$p>2$. 我们为这两种公式提供了几种有效且可扩展的算法，并提供了合成数据的数值结果，表明它们的收敛特性。最后，我们对 MNIST、FashionMNIST 和 EMNIST 数据集进行了广泛的数值实验，这些实验说明了p -Laplacian公式在几乎没有标签的半监督学习中的有效性。特别是，我们发现 Lipschitz 学习 ($p=\infty$) 在k -NN 图上的标签很少的情况下表现良好，这通过实验验证了我们的理论发现，即 Lipschitz 学习保留了有关k -NN 图上的数据分布（未标记数据）的信息。