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We relate function class complexity to structure in the function domain. This facilitates risk analysis relative to cluster structure in the input space which is particularly effective in semi-supervised learning. In particular we quantify the complexity of function classes defined over a graph in t ...

We continue our study of online prediction of the labelling of a graph. We show a fundamental limitation of Laplacian-based algorithms: if the graph has a large diameter then the number of mistakes made by such algorithms may be proportional to the square root of the number of vertices, even when ta ...