Scalings in which the graph Laplacian approaches a differential operator in the large graph limit are used to develop understanding of a number of algorithms for semi-supervised learning; in particular, the probit algorithm, level set and kriging methods. Both optimization and Bayesian approaches are considered, based around a regularizing quadratic form found from an affine transformation of the Laplacian, raised to a possibly fractional, exponent. Conditions on the parameters defining this quadratic form are identified under which well-defined limiting continuum analogues of the optimization and Bayesian semi-supervised learning problems may be found, thereby shedding light on the design of algorithms in the large graph setting. The large graph limits of the optimization formulations are tackled through Γ-convergence, using the recently introduced $T{L}^p$ metric. The small labeling noise limits of the Bayesian formulations are also identified, and contrasted with pre-existing harmonic function approaches to the problem.

图拉普拉斯算子在大图限制内逼近微分算子的标度用于发展对半监督学习的多种算法的理解。特别是概率算法，水平集和克里金法。基于从拉普拉斯仿射变换中发现的正则化二次形式的基础上，考虑了优化方法和贝叶斯方法，并提出了可能是分数形式的指数。确定定义该二次形式的参数的条件，在该条件下可以找到优化的定义明确的极限连续类比和贝叶斯半监督学习问题，从而为大型图形设置中的算法设计提供了启示。优化公式的较大图形限制可通过Γ收敛解决，$Ť{\mathrm{大号}}^p$指标。贝叶斯公式的小标签噪声极限也被确定，并且与该问题的预先存在的谐波函数方法形成对比。