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February 24 @ 4:00 pm - 5:00 pm
Title: Graph-based approximation of Matérn Gaussian fields.
Presenter: Ruiyi Yang (University of Chicago)
Abstract: Matérn Gaussian fields (MGF) have been popular modeling choices as priors in many aspects of Bayesian statistics. In this talk we will discuss a generalization of MGF to manifolds and graphs. In the first part, we formalize the definition of MGF on manifolds by exploiting the stochastic partial differential equation representation of the usual MGF on Euclidean domains. Sparse approximation based on a graph discretization is introduced, which further motivates the construction of MGF on graphs. We will demonstrate their applications in regression and classification problems. Building on the theory of spectral convergence of graph Laplacians, we establish a convergence analysis of graph MGF towards their continuum counterparts.
In the second part, we study in detail the Bayesian regression and classification problems in a semi-supervised framework, where a large number of unlabeled data are provided apart from the labeled ones. We adopt a graph-based approach as above and study quantitatively the benefit of unlabeled data through an analysis of the corresponding posterior contraction rates. Building on the convergence analysis in the first part, we show that the graph posterior contracts around the underlying truth at the minimax optimal rate (up to logarithmic factors) provided that sufficiently many unlabeled data are given.