Uncertainty quantification in graph-based classification of high dimensional data
This work addresses the need for uncertainty quantification in classification for applications where reliability is critical, though it is incremental as it builds upon and unifies prior graph-based methods.
The paper tackles binary classification of high-dimensional data by introducing Bayesian models that provide uncertainty measures via posterior distributions, all based on graph-based semi-supervised learning. It unifies existing methods, generalizes approaches like probit and level-set to this setting, and demonstrates classification accuracy and uncertainty quantification through numerical experiments on standard datasets.
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 in the graph-based setting, generalize the level-set method for Bayesian inverse problems to the classification setting, and generalize the Ginzburg-Landau optimization-based classifier to a Bayesian setting; we also show that the probit and level set approaches are natural relaxations of the harmonic function approach introduced in [Zhu et al 2003]. 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.