Distance metric learning under covariate shift

Learning distance metrics is a fundamental problem in machine learning. Previous distance-metric learning research assumes that the training and test data are drawn from the same distribution, which may be violated in practical applications. When the distributions differ, a situation referred to as covariate shift, the metric learned from training data may not work well on the test data. In this case the metric is said to be inconsistent. In this paper, we address this problem by proposing a novel metric learning framework known as consistent distance metric learning (CDML), which solves the problem under covariate shift situations. We theoretically analyze the conditions when the metrics learned under covariate shift are consistent. Based on the analysis, a convex optimization problem is proposed to deal with the CDML problem. An importance sampling method is proposed for metric learning and two importance weighting strategies are proposed and compared in this work. Experiments are carried out on synthetic and real world datasets to show the effectiveness of the proposed method.