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Abstract

This chapter presents a systemic framework for learning from a finite set represented as a graph. Discrete analogues are developed here of a number of differential operators, and then a discrete analogue of classical regularization theory is constructed based on those discrete differential operators. The graph Laplacian-based approaches are special cases of this general discrete regularization framework. More importantly, new approaches based on other different differential operators are derived as well. A variety of approaches for learning from finite sets has been proposed from different motivations and for different problems. In most of those approaches, a finite set is modeled as a graph, in which the edges encode pairwise relationships among the objects in the set. Consequently many concepts and methods from graph theory are applied, in particular, graph Laplacians.