Uncertainty Quantification in the Classification of High Dimensional Data

Classification of high dimensional data finds wide-ranging applications. In many of these applications equipping the resulting classification with a measure of uncertainty may be as important as the classification itself. In this paper we introduce, develop algorithms for, and investigate the properties of, a variety of Bayesian models for the task of binary classification; via the posterior distribution on the classification labels, these methods automatically give measures of uncertainty. The methods are all based around the graph formulation of semi-supervised learning. We provide a unified framework which brings together a variety of methods which have been introduced in different communities within the mathematical sciences. We study probit classification [50] in the graph-based setting, generalize the level-set method for Bayesian inverse problems [27] to the classification setting, and generalize the Ginzburg-Landau optimization-based classifier [7, 46] to a Bayesian setting; we also show that the probit and level set approaches are natural relaxations of the harmonic function approach introduced in [56]. We introduce efficient numerical methods, suited to large data-sets, for both MCMC-based sampling as well as gradient-based MAP estimation. Through numerical experiments we study classification accuracy and uncertainty quantification for our models; these experiments showcase a suite of datasets commonly used to evaluate graph-based semi-supervised learning algorithms.