Abstract: MR(Manifold Regularization) provides a powerful framework for semi-supervised classification using labeled and unlabeled datasets.Under the assumption of manifold,it enforces that similar instances should have similar classification results on the sample graph.It is notable that the core of MR lies in the pairwise smoothing on the sample graph,where smoothing constraints are applied to all instance pairs,treating each pair of instances as a whole.However,the smoothness can be point-to-point in essence,meaning that smoothness should be “everywhere” to correlate the behavior of each point or instance with that of its neighboring points.Therefore,this paper proposes a novel semi-supervised learning algorithm based on pointwise manifold and uniform regularity constraints(URC-PW-MR),which achieves semi-supervised learning by constraining individual local instances and introducing a fusion consistency regularization.This approach not only contains the pointwise nature of smoothness but also introduces the importance of individual instance by considering each instance rather than pairs of instances.The significance can be quantitatively characterized through factors such as local density.URC-PW-MR proposes a novel manifold smoothness realization approach that achieves semi-supervised learning through dual constraints:individual local instance regularization and fusion consistency regularization.Empirical evaluations demonstrate that URC-PW-MR exhibits more refined performance characteristics compared with conventional MR frameworks.

Key words: Manifold regularization, Pointwise manifold regularization, Uniform regularity constraints, Semi-supervised learning