Global and local estimation of low-rank random graphs
Random graph models have been a heated topic in statistics and machine learning, as well as a broad range of application areas. In this talk I will give two perspectives on the estimation task of low-rank random graphs. Specifically, I will focus on estimating the latent positions in random dot product graphs. The first component of the talk focuses on the global estimation task. The minimax lower bound for global estimation of the latent positions is established, and this minimax lower bound is achieved by a Bayes procedure, referred to as the posterior spectral embedding. The second component of the talk addresses the local estimation task. We define local efficiency in estimating each individual latent position, propose a novel one-step estimator that takes advantage of the curvature information of the likelihood function (i.e., derivatives information) of the graph model, and show that this estimator is locally efficient. The previously widely adopted adjacency spectral embedding is proven to be locally inefficient due to the ignorance of the curvature information of the likelihood function. Simulation examples and the analysis of a real-world Wikipedia graph dataset are provided to demonstrate the usefulness of the proposed methods.