Semi-supervised learning with manifold regularization is a classical framework for jointly learning from both labeled and unlabeled data, where the key requirement is that the support of the unknown marginal distribution has the geometric structure of a Riemannian manifold. Typically, the Laplace-Beltrami operator-based manifold regularization can be approximated empirically by the Laplacian regularization associated with the entire training data and its corresponding graph Laplacian matrix. However, the graph Laplacian matrix depends heavily on the prespecified similarity metric and may lead to inappropriate penalties when dealing with redundant or noisy input variables. To address the above issues, this paper proposes a new \textit{Semi-Supervised Meta Additive Model (S$^2$MAM) based on a bilevel optimization scheme that automatically identifies informative variables, updates the similarity matrix, and simultaneously achieves interpretable predictions. Theoretical guarantees are provided for S$^2$MAM, including the computing convergence and the statistical generalization bound. Experimental assessments across 4 synthetic and 12 real-world datasets, with varying levels and categories of corruption, validate the robustness and interpretability of the proposed approach.

翻译：基于流形正则化的半监督学习是一种利用标记和未标记数据共同学习的经典框架，其关键假设在于未知边际分布的支持集具有黎曼流形的几何结构。通常，基于Laplace-Beltrami算子的流形正则化可通过与全部训练数据及其对应的图拉普拉斯矩阵相关的拉普拉斯正则化进行经验近似。然而，图拉普拉斯矩阵高度依赖于预先指定的相似性度量，在处理冗余或噪声输入变量时可能导致不恰当的惩罚。针对上述问题，本文提出一种新的基于双层优化策略的半监督元可加模型（S$^2$MAM），该模型可自动识别信息变量、更新相似性矩阵，并同时实现可解释的预测。本文为S$^2$MAM提供了理论保障，包括计算收敛性和统计泛化界。在4个合成数据集和12个真实世界数据集上进行的实验评估（涵盖不同程度和类型的噪声干扰）验证了所提方法的鲁棒性和可解释性。