Various graph-based algorithms for semi-supervised learning have been proposed in the recent literature. They rely on the idea of building a graph whose nodes are data points (labeled and unlabeled) and edges represent similarities between points. Known labels are used to propagate information through the graph in order to label all nodes. In this chapter, we show how these different algorithms can be cast into a common framework where one minimizes a quadratic cost criterion whose closed-form solution is found by solving a linear system of size n (total number of data points). The cost criterion naturally leads to an extension of such algorithms to the inductive setting, where one obtains test samples one at a time: the derived induction formula can be evaluated in O(n) time, which is much more efficient than solving again exactly the linear system (which in general costs O(kn2) time for a sparse graph where each data point has k neighbors). We also use this inductive formula to show that when the similarity between points satisfies a locality property, then the algorithms are plagued by the curse of dimensionality, with respect to the dimensionality of an underlying manifold.
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