Abstract
Learning Mahalanobis metric spaces is an important problem that has found numerous applications. Several algorithms have been designed for this problem, including Information Theoretic Metric Learning (ITML) [Davis et al. 2007] and Large Margin Nearest Neighbor (LMNN) classification [Weinberger and Saul 2009]. We study the problem of learning a Mahalanobis metric space in the presence of adversarial label noise. To that end, we consider a formulation of Mahalanobis metric learning as an optimization problem, where the objective is to minimize the number of violated similarity/dissimilarity constraints. We show that for any fixed ambient dimension, there exists a fully polynomial-time approximation scheme (FPTAS) with nearly-linear running time. This result is obtained using tools from the theory of linear programming in low dimensions. As a consequence, we obtain a fully-parallelizable algorithm that recovers a nearly-optimal metric space, even when a small fraction of the labels is co