Universally consistent vertex classification for latent positions graphs

TL;DR

This paper demonstrates that eigen-decomposition of adjacency matrices can consistently estimate feature maps in latent position graphs, enabling universally consistent vertex classification with increasing data and classifier complexity.

Contribution

It introduces a method for consistent feature map estimation and vertex classification in latent position graphs using eigen-decomposition and universal kernels.

Findings

Eigen-decomposition yields consistent feature maps for latent position graphs.

A universal classifier converges to the Bayes optimal error as data increases.

The approach applies to graphs with positive definite link functions and universal kernels.

Abstract

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 $\kappa$, 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 $\kappa$ 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 $\varphi$-risk for some convex surrogate $\varphi$ 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.