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Abstract:
Many real-world machine learning problems are situated on finite discrete sets,
including dimensionality reduction, clustering, and transductive inference. 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
adopted. In particular, the graph Laplacian is widely used.
In this chapter we present a systemic framework for learning from a finite set
represented as a graph. We develop discrete analogues of a number of differential
operators, and then construct a discrete analogue of classical regularization theory
based on those discrete differential operators. The graph Laplacian based approaches
are special cases of this general discrete regularization framework. An important
thing implied in this framework is that we have a wide choices of regularization on
graph in addition to the widely-used graph Laplacian based one.
