Distance Metric Learning by Optimization on the Stiefel Manifold

Abstract

Distance metric learning has proven to be very successful in various problem domains. Most techniques learn a global metric in the form of a n × n symmetric positive semidefinite (PSD) Mahalanobis distance matrix, which has O(n 2 ) unknowns. The PSD constraint makes solving the metric learning problem even harder making it computationally intractable for high dimensions. In this work, we propose a flexible formulation that can employ different regularization functions, while implicitly maintaining the positive semidefiniteness constraint. We achieve this by eigendecomposition of the rank p Mahalanobis distance matrix followed by a joint optimization on the Stiefel manifold S n,p and the positive orthant R p + . The resulting nonconvex optimization problem is solved by employing an alternating strategy. We use a recently proposed projection free approach for efficient optimization over the Stiefel manifold. Even though the problem is nonconvex, we empirically show competitive classification accuracy on UCI and USPS digits datasets.