Making metric learning algorithms invariant to transformations using a projection metric on Grassmann manifolds

The requirement for suitable ways to measure the distance or similarity between data is omnipresent in machine learning, pattern recognition and data mining, but extracting such good metrics for particular problems is in general challenging. This has led to the emergence of metric learning ideas, which intend to automatically learn a distance function tuned to a specific task. In many tasks and data types, there are natural transformations to which the classification result should be invariant or insensitive. This demand and its implications are essential in many machine learning applications, and insensitivity to image transformations was in the first place achieved by using invariant feature vectors. In this paper, a new representation model on Grassmann manifolds for data points and a novel method for learning a Mahalanobis metric which uses the geodesic distance on Grassmann manifolds are proposed. In fact, we use an appropriate geodesic distance metric on the Grassmann manifolds, called projection metric, for measuring primary similarities between the new representations of the data points. This makes learning of the Mahalanobis metric invariant to similarity transforms and intensity changes, and therefore improve the performance. Experiments on face and handwritten digit datasets demonstrate that our proposed method yields performance improvements in a state-of-the-art metric learning algorithm.