A scalable dual approach to semidefinite metric learning

Abstract

Distance metric learning plays an important role in many vision problems. Previous work of quadratic Mahalanobis metric learning usually needs to solve a semidefinite programming (SDP) problem. A standard interior-point SDP solver has a complexity of O(D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6.5</sup> ) (with D the dimension of input data), and can only solve problems up to a few thousand variables. Since the number of variables is D(D + l)/2, this corresponds to a limit around D <; 100. This high complexity hampers the application of metric learning to high-dimensional problems. In this work, we propose a very efficient approach to this metric learning problem. We formulate a Lagrange dual approach which is much simpler to optimize, and we can solve much larger Mahalanobis metric learning problems. Roughly, the proposed approach has a time complexity of O(t · D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) with t ≈ 20 ~ 30 for most problems in our experiments. The proposed algorithm is scalable and easy to implement. Experiments on various datasets show its similar accuracy compared with state-of-the-art. We also demonstrate that this idea may also be able to be applied to other SDP problems such as maximum variance unfolding.