The Annals of Statistics

Universally consistent vertex classification for latent positions graphs

Minh Tang, Daniel L. Sussman, and Carey E. Priebe

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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 $F$.

Article information

Ann. Statist., Volume 41, Number 3 (2013), 1406-1430.

First available in Project Euclid: 1 August 2013

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Zentralblatt MATH identifier

Primary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]
Secondary: 62C12: Empirical decision procedures; empirical Bayes procedures 62G20: Asymptotic properties

Classification latent space model convergence of eigenvectors Bayes-risk consistency convex cost function


Tang, Minh; Sussman, Daniel L.; Priebe, Carey E. Universally consistent vertex classification for latent positions graphs. Ann. Statist. 41 (2013), no. 3, 1406--1430. doi:10.1214/13-AOS1112.

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