Pointwise manifold regularization for semi-supervised learning

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.