Manifold regularization (MR) provides a powerful framework for semi-supervised classification using both the labeled and unlabeled data. It constrains that similar instances over the manifold graph should share similar classification outputs according to the manifold assumption. It is easily noted that MR is built on the pairwise smoothness over the manifold graph, i.e., the smoothness constraint is implemented over all instance pairs and actually considers each instance pair as a single operand. However, the smoothness can be pointwise in nature, that is, the smoothness shall inherently occur “everywhere” to relate the behavior of each point or instance to that of its close neighbors. Thus in this paper, we attempt to develop a pointwise MR (PW_MR for short) for semi-supervised learning through constraining on individual local instances. In this way, the pointwise nature of smoothness is preserved, and moreover, by considering individual instances rather than instance pairs, the importance or contribution of individual instances can be introduced. Such importance can be described by the confidence for correct prediction, or the local density, for example. PW_MR provides a different way for implementing manifold smoothness. Finally, empirical results show the competitiveness of PW_MR compared to pairwise MR.

中文翻译：

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半监督学习的点状流形正则化

歧管正则化（MR）为使用标记和未标记数据的半监督分类提供了强大的框架。它限制了流形图上的相似实例应根据流形假设共享相似的分类输出。容易注意到，MR建立在流形图上的成对平滑度上，即，平滑度约束是在所有实例对上实现的，实际上将每个实例对视为单个操作数。但是，平滑度本质上可以是逐点的，也就是说，平滑度将固有地在“任何地方”发生，以将每个点或实例的行为与其相邻近邻的行为联系起来。因此，在本文中，我们尝试通过约束单个局部实例来开发半监督学习的逐点MR（简称PW_MR）。以这种方式，保持了平滑的点状性质，此外，通过考虑单个实例而不是实例对，可以引入单个实例的重要性或贡献。可以通过例如正确预测的置信度或局部密度来描述这种重要性。PW_MR提供了实现流形平滑度的另一种方法。最后，实证结果表明PW_MR与成对MR相比具有竞争力。PW_MR提供了实现流形平滑度的另一种方法。最后，实证结果表明PW_MR与成对MR相比具有竞争力。PW_MR提供了实现流形平滑度的另一种方法。最后，实证结果表明PW_MR与成对MR相比具有竞争力。