Universally consistent vertex classification for latent positions graphs

In this work we show that, using the eigen-decomposition of the adjacency matrix, we can consistently estimate feature maps for latent position graphs with positive definite link function $κ$, provided that the latent positions are i.i.d. from some distribution F. We then consider the exploitation task of vertex classification where the link function $κ$ belongs to the class of universal kernels and class labels are observed for a number of vertices tending to infinity and that the remaining vertices are to be classified. We show that minimization of the empirical $φ$-risk for some convex surrogate $φ$ of 0-1 loss over a class of linear classifiers with increasing complexities yields a universally consistent classifier, that is, a classification rule with error converging to Bayes optimal for any distribution F.

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