Abstract

This paper introduces a regularization method to ex-plicitly control the rank of a learned symmetric positivesemidefinite distance matrix in distance metric learning. Tothis end, we propose to incorporate in the objective functiona linear regularization term that minimizes the k smallesteigenvalues of the distance matrix. It is equivalent to min-imizing the trace of the product of the distance matrix witha matrix in the convex hull of rank-k projection matrices,called a Fantope. Based on this new regularization method,we derive an optimization scheme to efficiently learn the distance matrix. We demonstrate the effectiveness of the method on synthetic and challenging real datasets of face verification and image classification with relative attributes,on which our method outperforms state-of-the-art metric learning algorithms.