Robust and effective metric learning using capped trace norm

Metric learning aims at automatically learning a metric from pair or triplet based constraints in data, and it can be potentially beneficial whenever the notion of metric between instances plays a nontrivial role. In Mahalanobis distance metric learning, distance matrix M is in symmetric positive semi-definite cone, and in order to avoid overfitting and to learn a better Mahalanobis distance from weakly supervised constraints, the low-rank regularization has been often imposed on matrixM to learn the correlations between features and samples. As the approximations of the rank minimization function, the trace norm and Fantope have been utilized to regularize the metric learning objectives and achieve good performance. However, these low-rank regularization models are either not tight enough to approximate rank minimization or time-consuming to tune an optimal rank. In this paper, we introduce a novel metric learning model using the capped trace norm based regularization, which uses a singular value threshold to constraint the metric matrixM as low-rank explicitly such that the rank of matrix M is stable when the large singular values vary. The capped trace norm regularization can also be viewed as the adaptive Fantope regularization. We minimize singular values which are less than threshold value and the rank of M is not necessary to be k, thus our method is more stable and applicable in practice when we do not know the optimal rank of matrix M. We derive an efficient optimization algorithm to solve the proposed new model and the algorithm convergence proof is also provided in this paper. We evaluate our method on a variety of challenging benchmarks, such as LFW and Pubfig datasets. Face verification experiments are performed and results show that our method consistently outperforms the state-of-the-art metric learning algorithms.